Recovery and continuity of surfaces with assigned director field

Abstract

We show that surfaces with assigned director field immersed in the three-dimensional Euclidean space can be defined intrinsically by four tensor fields, of which two are of order two, one of order one, and one of order zero. We then show how a surface and its assigned director field can be reconstructed from these four tensors and prove that the reconstruction operator is continuous between ad hoc functional spaces with as little regularity as possible. These results have applications in the Cosserat theory of nonlinearly elastic shells, the strain energy of which are defined precisely in terms of the four tensor fields defined in this paper.

Keywords

Differential geometry surface theory Gauss–Codazzi equations Cosserat shell rigging vector fields intrinsic theory of shells

1. Introduction

Our study of surfaces with assigned director field is motivated by the theory of single-director Cosserat shells, where shells are defined as three-dimensional bodies of the form

Ω (φ, d) : = {φ (y) + x_{3} d (y); y \in \bar{ω}, x_{3} \in [- ε, ε]},

where ω is a domain in $ℝ^{2}$ , $φ : \bar{ω} \to ℝ^{3}$ is a smooth enough embedding, $d : \bar{ω} \to ℝ^{3}$ is a smooth enough unit vector field transversal to the surface $S : = φ (\bar{ω})$ , and $ε > 0$ is a small enough parameter. In shell theory, S represents the middle surface of an elastic shell, ε represents the half-thickness of the shell, and d represents the direction of the transversal elastic fibers of the shell. The particular case where d is normal to the surface S at every point corresponds to the Kirchhoff–Love theory of shells, so this paper can be seen as a generalization of previous works about the latter shells by Ciarlet et al. [1 –7].

According to the theory of single-director Cosserat shells, the strain energy density per unit area of S associated with the deformation of a shell from a reference configuration $Ω (φ, d)$ to a new configuration $Ω (\tilde{φ}, \tilde{d})$ is defined by (see, e.g., Naghdi [8])

\frac{ε}{8} ∥ ({\tilde{a}}_{αβ} - a_{αβ}) ∥_{λ, μ}^{2} + \frac{ε^{3}}{6} ∥ ({\tilde{k}}_{αβ} - k_{αβ}) ∥_{λ, μ}^{2} + ε ∥ ({\tilde{d}}_{α} - d_{α}) ∥_{μ}^{2},

where $(a_{αβ}, k_{αβ}, d_{α})$ are the tensor fields associated with $(φ, d)$ by the relations

\begin{matrix} a_{αβ} & : = \partial_{α} φ \cdot \partial_{β} φ, \\ k_{αβ} & : = \frac{1}{2} (\partial_{α} φ \cdot \partial_{β} d + \partial_{α} d \cdot \partial_{β} φ), \\ d_{α} & : = \partial_{α} φ \cdot d, \end{matrix}

and similar definitions hold for the tensors $({\tilde{a}}_{αβ}, {\tilde{k}}_{αβ}, {\tilde{d}}_{α})$ .

The norms appearing in the definition of the above strain energy are defined, for each symmetric tensor field $(s_{αβ})$ and for each vector field $(s_{α})$ , by

\begin{matrix} ∥ (s_{αβ}) ∥_{λ, μ}^{2} & : = a^{αβστ} s_{στ} s_{αβ}, \\ ∥ (s_{α}) ∥_{μ}^{2} & : = μ a^{αβ} s_{α} s_{β}, \end{matrix}

where $λ \geq 0$ and $μ > 0$ denote the Lamé constants of the elastic material constituting the shell, $(a^{αβ})$ denotes the inverse matrix of $(a_{αβ})$ , and

a^{αβστ} : = \frac{4 λμ}{λ + 2 μ} a^{αβ} a^{στ} + 2 μ (a^{ασ} a^{βτ} + a^{ατ} a^{βσ})

denotes the two-dimensional elasticity tensor associated with the elastic material constituting the shell.

The objective of this paper is to investigate the properties of the mapping

(φ, d) \to (a_{αβ}, k_{αβ}, d_{α}),

such as identifying its domain and range and studying its continuity and the continuity of its inverse mapping between appropriate function spaces, properties which are essential in shell theory.

We will show, in particular, that this mapping is well defined between the spaces $W_{loc}^{2, p} \times W_{loc}^{1, p}$ and $W_{loc}^{1, p} \times L_{loc}^{p} \times W_{loc}^{1, p}$ , $p > 2$ (Theorem 1), and we establish compatibility conditions satisfied by all elements in the range of this mapping (Theorems 2 and 3).

Furthermore, under the assumption that ω is simply connected, we show that the functions $(φ, d)$ defining the shell can be reconstructed, up to a proper isometry of the Euclidean space, from the tensors $(a_{αβ}, k_{αβ}, d_{α})$ provided they satisfy the compatibility conditions mentioned above (Theorem 4), thus establishing a generalization of the fundamental theorem of surface theory. Finally, we prove that the reconstruction mapping in Theorem 4 is continuous between appropriate topological spaces (Theorem 7).

In fact, we will consider the more general case where $n : = | d |^{2}$ is not necessary equal to one and deduce the properties of the above mapping simply by letting $n = 1$ in our more general results. This strategy is justified by Le Dret and Raoult’s analysis of shell models with one director field (see [9]) and also by possible applications of our results in Relativity theory, where surfaces with a rigging vector field d are used but with d not necessarily of norm one (see, e.g., LeFloch et al. [10]).

The paper is self-contained, in the sense that all its proofs, with the exception of implication (ii) in the proof of Theorem 5, on purpose, are made independent of those theorems in the classical theory of surfaces that could otherwise be used instead of some arguments in our proofs. In this way, the statements and proofs of these classical theorems can be recovered from their generalizations in this paper simply by letting $d_{α} = 0$ and $n = 1$ .

Another advantage of our approach of giving independent proofs, as opposed to using the classical theory of surfaces to deduce these theorems by appropriate change of variables and unknowns, is the realization that they hold in fact under weaker regularity assumptions on the domain ω, namely, for domains satisfying the uniform interior cone property instead of domains with Lipschitz-continuous boundary. Otherwise, it turns out that the reconstruction of a surface with assigned director field (see Theorem 7) holds in functional spaces that are similar to those used in the reconstruction of a surface, which suggests that using a director field d instead of the normal vector field $a_{3}$ might not be sufficient to establish existence theorems for nonlinear shell models of Koiter’s type.

The results established in this paper can be generalized to hypersurfaces in the Euclidean space $ℝ^{d}$ , $d > 3$ , by adapting the functional framework used in Malin and Mardare [11] to hypersurfaces with director field, which entails replacing the vector product ∧ in $ℝ^{3}$ by the exterior product in $ℝ^{d}$ . They can be furthermore generalized to submanifolds in $ℝ^{d}$ of arbitrary codimension $q \geq 1$ by adapting the functional framework used in Szopos [12] to submanifolds with q director fields. In this case, additional structure is required in the form of the equivalent for submanifolds with director fields of the normal connection and of the Gauss–Ricci–Codazzi equations for classical submanifolds.

2. Notation and definitions

This section specifies the notation and definitions used in this paper.

Greek indices vary in the set ${1, 2}$ , while Latin indices vary in the set ${1, 2, 3}$ . The summation convention for repeated indices is used in conjunction with this rule. Any relation featuring Greek or Latin indices holds for all values of the indices in the above ranges; e.g., relations (1) and (2) in the next section hold for all $α, β \in {1, 2}$ .

Vector and matrix fields are denoted by boldface letters, while scalar fields (which are simply real-valued functions) are denoted by simple letters.

The three-dimensional space $ℝ^{3}$ is equipped with the scalar product ⋅ and the vector product ∧ respectively defined by $a \cdot b : = \sum_{i = 1}^{3} a_{i} b_{i}$ and $a \land b : = (a_{2} b_{3} - a_{3} b_{2}, a_{3} b_{1} - a_{1} b_{3}, a_{1} b_{2} - a_{2} b_{1})$ for all vectors $a = (a_{1}, a_{2}, a_{3}) \in ℝ^{3}$ and $b = (b_{1}, b_{2}, b_{3}) \in ℝ^{3}$ . The Euclidean norm in the same space is denoted and defined by $| a | : = {(a \cdot a)}^{1 ∕ 2}$ for all vectors $a \in ℝ^{3}$ .

The space of all $k \times ℓ$ matrices with real coefficients is denoted $ℝ^{k \times ℓ}$ . It is equipped with the Frobenius norm, which is denoted and defined by $| A | : = {(Tr (A^{T} A))}^{1 ∕ 2}$ for all $A \in ℝ^{k \times ℓ}$ , where $Tr : ℝ^{ℓ \times ℓ} \to ℝ$ denotes the trace operator. The matrix with $a_{ij}$ , respectively, $a_{j}^{i}$ , as its component at the i-th row and j-th column is denoted $(a_{ij})$ , respectively, $(a_{j}^{i})$ . The transpose of a matrix $A = (a_{ij})$ is denoted and defined by $A^{T} = (a_{ji})$ . An orthogonal matrix is a square real matrix A that satisfies $A^{T} A = A A^{T} = I$ , where I denotes the identity matrix of the same order as A . A special orthogonal matrix is an orthogonal matrix with positive determinant.

In all that follows, ω denotes a connected and open subset of the two-dimensional space $ℝ^{2}$ with Cartesian coordinates $y = (y_{α})$ . Partial differential operators applied to scalar, vector, or tensor, fields defined over ω are denoted $\partial_{α} : = \partial ∕ \partial y_{α}$ and $\partial_{αβ} : = \partial^{2} ∕ \partial y_{α} \partial y_{β}$ . The space of distributions over ω is denoted $D^{'} (ω)$ .

Spaces of scalar fields (real-valued functions) are denoted by specifying only their domain (their range being always ℝ), while spaces of vector or matrix fields are denoted by specifying their domain and range. For instance, $C^{0} (ω)$ denotes the space of continuous real-valued functions defined over the set ω, while $C^{0} (ω; ℝ^{3})$ designates the space of vector fields $v = {(v_{i})}_{i = 1}^{3} : ω \to ℝ^{3}$ such that $v_{i} \in C^{0} (ω)$ for all $i \in {1, 2, 3}$ . A function is of class $C^{k}$ in ω, $k \in ℕ^{*}$ , if it is k times differentiable in ω and all its partial derivatives up to order k are continuous in ω.

The space of all bounded continuous functions $f : ω \to ℝ$ is denoted $C_{b}^{0} (ω)$ . It is equipped with the norm denoted and defined by

∥ f ∥_{L^{\infty} (ω)} : = \sup_{y \in ω} | f (y) | for all f \in C_{b}^{0} (ω) .

The notation $C^{0} (\bar{ω})$ denotes the set of all continuous functions $f : ω \to ℝ$ that possess continuous extensions to the closure $\bar{ω}$ of ω. If ω is bounded, then $C^{0} (\bar{ω})$ is a subspace $C_{b}^{0} (ω)$ , equipped with the induced norm $∥ \cdot ∥_{L^{\infty} (ω)}$ .

For each real number $p \geq 1$ and each integer $k \geq 1$ , $L^{p} (ω) = W^{0, p} (ω)$ and $W^{k, p} (ω)$ , respectively, denote the Lebesgue and Sobolev spaces equipped with their usual norms, respectively, denoted $∥ \cdot ∥_{L^{p} (ω)}$ and $∥ \cdot ∥_{W^{k, p} (ω)}$ . When $p > 2$ and $k \geq 1$ , the space $W^{k, p} (ω)$ in an algebra and is identified with the subspace of $C^{k - 1} (ω)$ formed by the (unique) continuous representatives of its elements (which are equivalence classes of functions with respect to the equality almost everywhere).

Lebesgue spaces of vector and matrix fields over ω are respectively by denoted $L^{p} (ω; ℝ^{m})$ and $L^{p} (ω; ℝ^{m \times ℓ})$ . Their norms are defined in terms of the norms of the spaces $L^{p} (ω)$ and $W^{k, p} (ω)$ by

∥ v ∥_{L^{p} (ω)} : = ∥ | v (x) | ∥_{L^{p} (ω)} for all v \in L^{p} (ω; ℝ^{m}),

where |⋅| denotes the Euclidean norm, and

∥ F ∥_{L^{p} (ω)} : = ∥ | F (x) | ∥_{L^{p} (ω)} for all F \in L^{p} (ω; ℝ^{m \times ℓ}),

where |⋅| denotes the Frobenius norm. Note that these norms are invariant under rotations, in the sense that, if $R_{0}$ is a $m \times m$ orthogonal matrix, and $Q_{0}$ is a $ℓ \times ℓ$ orthogonal matrix, then

∥ R_{0} v ∥_{L^{p} (ω)} = ∥ v ∥_{L^{p} (ω)} and ∥ R_{0} F Q_{0} ∥_{L^{p} (ω)} = ∥ F ∥_{L^{p} (ω)} .

The space $W^{1, p} (ω; ℝ^{3})$ of all vector fields with components in the Sobolev space $W^{1, p} (ω)$ is equipped with the norm

\begin{matrix} ∥ φ ∥_{W^{1, p} (ω)} & : = {(∥ φ ∥_{L^{p} (ω)}^{p} + \sum_{α} ∥ \partial_{α} φ ∥_{L^{p} (ω)}^{p})}^{1 ∕ p} & if p < \infty, \\ : = \max (∥ φ ∥_{L^{p} (ω)}, ∥ \partial_{1} φ ∥_{L^{p} (ω)}, ∥ \partial_{2} φ ∥_{L^{p} (ω)}) & if p = \infty, \end{matrix}

for all $φ = (φ_{i}) \in W^{1, p} (ω; ℝ^{3})$ . Similar definitions hold for the norm of the spaces $W^{1, p} (ω; ℝ^{3 \times 3})$ and $W^{2, p} (ω; ℝ^{3})$ .

The notations $L_{loc}^{p} (ω)$ and $W_{loc}^{1, p} (ω)$ denote the spaces of measurable functions from ω into ℝ whose restrictions to any subset $ω_{c} ⋐ ω$ belong to $L^{p} (ω_{c})$ and $W^{1, p} (ω_{c})$ , respectively. The notation $ω_{c} ⋐ ω$ means that $ω_{c}$ is an open and bounded subset of $ℝ^{2}$ whose closure is contained in ω. These spaces are equipped with their natural Fréchet topologies.

An immersion from ω into $ℝ^{3}$ is a function $φ \in C^{1} (ω; ℝ^{3})$ whose partial derivatives of order one, which form a family of two vector fields $\partial_{α} φ$ defined over ω with values in $ℝ^{3}$ , are linearly independent at each point of ω.

A notation such as $d_{i} = d_{i} (φ, d)$ means that $d_{i}$ depends on φ and d , and only on them.

3. The geometry of a surface with assigned director field

In this section, we recall the classical notions describing the geometry of a surface, such as its first and second fundamental forms $(a_{αβ})$ and $(b_{αβ})$ , and the associated Levi–Civita connection defined by means of the Christoffel symbols of the second kind $(Γ_{αβ}^{τ})$ , then we introduce new notions specific to the geometry of a surface with assigned director field, such that the tensor field $(k_{αβ})$ and the coefficients $C_{αj}^{i}$ of the moving frame equations associated with a director field d transversal to the surface at every point of the surface. We will prove that the functions $C_{αj}^{i}$ depend in fact only on the tensor fields $(a_{αβ})$ and $(k_{αβ})$ and on two fields $(d_{α})$ and n characterizing the director field d . Then, we will establish ad hoc compatibility conditions for these tensor and director fields that generalize the classical Gauss–Codazzi equations for surfaces in $ℝ^{3}$ .

Given any immersion $φ : ω \to ℝ^{3}$ of class $C^{1}$ , let the vector fields $a_{i} : ω \to ℝ^{3}$ be defined by

a_{1} : = \partial_{1} φ, a_{2} : = \partial_{2} φ, a_{3} : = \frac{a_{1} \land a_{2}}{| a_{1} \land a_{2} |},

and let $a^{i} : ω \to ℝ^{3}$ be the unique vector fields defined by the relations

a^{i} \cdot a_{j} = δ_{j}^{i} in ω,

where $δ_{j}^{i}$ denotes the Kronecker symbol (i.e., $δ_{j}^{i} = 1$ if $i = j$ and $δ_{j}^{i} = 0$ otherwise).

Then, the two families ${a_{1} (y), a_{2} (y), a_{3} (y)}$ and ${a^{1} (y), a^{2} (y), a^{3} (y)}$ are two positively oriented bases in $ℝ^{3}$ for every $y \in ω$ , dual to each other, such that

a_{3} = a^{3}, a_{3} \cdot a_{α} = a_{3} \cdot a^{α} = 0, a_{3} \cdot a_{3} = 1 and a_{α} \cdot a^{β} = δ_{α}^{β} in ω .

Thus, for every $y \in ω$ , $a_{3} (y)$ is a unit vector perpendicular to the surface $S : = φ (ω)$ at the point $φ (y)$ , while the vectors $a_{α} (y)$ and $a^{α} (y)$ are tangent to the surface S at the same point. In fact, the two families ${a_{1} (y), a_{2} (y)}$ and ${a^{1} (y), a^{2} (y)}$ are two bases in the tangent plane to S at the point $φ (y)$ , dual to each other.

Given any immersion $φ : ω \to ℝ^{3}$ of class $C^{1}$ , the functions

a_{αβ} = a_{αβ} (φ) : = a_{α} \cdot a_{β}

(1)

and

a^{αβ} = a^{αβ} (φ) : = a^{α} \cdot a^{β}

(2)

respectively denote the covariant components and the contravariant components of the metric tensor associated with the immersion φ . Then, $(a_{αβ})$ and $(a^{αβ})$ are positive-definite symmetric matrix fields, inverse to each other, i.e.,

a^{ασ} a_{σβ} = a_{σβ} a^{ασ} = δ_{β}^{α} in ω,

where $δ_{β}^{α}$ denotes the Kronecker’s symbol.

For immersions φ of class $W_{loc}^{2, p} (ω)$ , $p > 2$ , the functions

Γ_{αβ}^{τ} = Γ_{αβ}^{τ} (φ) : = \frac{1}{2} a^{στ} (\partial_{α} a_{βσ} + \partial_{β} a_{ασ} - \partial_{σ} a_{αβ}),

(3)

which are well defined in the space $L_{loc}^{p} (ω)$ , denote the Christoffel symbols of second kind associated with the metric tensor $a_{αβ} a^{α} \otimes a^{β}$ . They satisfy the relations

Γ_{αβ}^{τ} = Γ_{βα}^{τ} = \partial_{α} a_{β} \cdot a^{τ} = \partial_{β} a_{α} \cdot a^{τ} = \partial_{αβ} φ \cdot a^{τ} .

For mappings $φ : ω \to ℝ^{3}$ of class $W_{loc}^{1, p} (ω)$ , $p \geq 2$ , such that the vector field $a_{3} : = (\partial_{1} φ \land \partial_{2} φ) ∕ | \partial_{1} φ \land \partial_{2} φ |$ is well defined and belongs to the space $W_{loc}^{1, p} (ω; ℝ^{3})$ , Ciarlet and Mardare [13, Lemma 3.2] proved that

a_{α} \cdot \partial_{β} a_{3} = a_{β} \cdot \partial_{α} a_{3} .

The functions

b_{αβ} = b_{αβ} (φ) : = - a_{α} \cdot \partial_{β} a_{3},

(4)

which then satisfy $b_{αβ} = b_{βα}$ in ω, denote the covariant components of the second fundamental form associated with the immersion φ , and the functions

b_{β}^{τ} : = b_{αβ} a^{ατ} and b^{τσ} : = b_{αβ} a^{ατ} a^{βσ}

(5)

respectively denote the mixed components and the contravariant components of the same second fundamental form. Note that $b^{τσ} = b^{στ}$ , but $b_{β}^{τ} \neq b_{τ}^{β}$ . If the immersion φ is of class $W_{loc}^{2, p} (ω)$ , $p > 2$ , then

b_{αβ} = \partial_{αβ} φ \cdot a_{3}

and the symmetry relations $b_{αβ} = b_{βα}$ are obvious in this case.

Finally, the above definitions of the Christoffel symbols and second fundament form imply that the vectors fields $a_{i}$ and $a^{i}$ satisfy the following Gauss and Weingarten equations

\begin{matrix} \partial_{α} a_{β} & = Γ_{αβ}^{σ} a_{σ} + b_{αβ} a_{3} & in ω, \\ \partial_{α} a_{3} & = - b_{α}^{σ} a_{σ} & in ω, \end{matrix}

(6)

which are equivalent to the system

\begin{matrix} \partial_{α} a^{β} & = - Γ_{ασ}^{β} a^{σ} + b_{α}^{β} a_{3} & in ω, \\ \partial_{α} a^{3} & = - b_{ασ} a^{σ} & in ω . \end{matrix}

(7)

We now describe the main features of the geometry of surfaces with assigned director fields.

Definition 1. A surface with assigned director field is a pair $(φ, d)$ formed by a mapping $φ : ω \to ℝ^{3}$ of class $C^{1} (ω)$ and a vector field $d : ω \to ℝ^{3}$ of class $C^{0} (ω)$ such that

(\partial_{1} φ \land \partial_{2} φ) \cdot d > 0 in ω .

Note that this definition implies that the three vector fields

a_{1} : = \partial_{1} φ, a_{2} : = \partial_{2} φ, d,

form a positively oriented basis in $ℝ^{3}$ at all points of ω. In particular, then, for all $y \in ω$ , the vector $d (y)$ is transversal to the surface S (i.e., $d (y)$ is not tangent to the surface S at $φ (y)$ ) and it is oriented towards the same side of S as the normal vector field $a_{3} : = \frac{a_{1} \land a_{2}}{| a_{1} \land a_{2} |}$ .

Remark 1. The assumption that $(\partial_{1} φ \land \partial_{2} φ) \cdot d > 0$ in ω is useful in shell theory to ensure that deformations of a shell defined in terms of the fields $(φ, d)$ preserve the orientation.

Remark 2. If a mapping φ of class $C^{1}$ and a vector field d of class $C^{0}$ satisfy

(\partial_{1} φ \land \partial_{2} φ) \cdot d > 0 at all points of ω,

then φ is an immersion from ω into $ℝ^{3}$ and the image $S : = φ (ω)$ is an oriented surface in $ℝ^{3}$ . Thus, a surface with director field is, in particular, an oriented surface.

Let

d_{i} = d_{i} (φ, d) : = d \cdot a_{i} and d^{i} = d^{i} (φ, d) : = d \cdot a^{i}

respectively denote the covariant components and the contravariant components of the vector field $d : ω \to ℝ^{3}$ . Then

d = d_{i} a^{i} = d^{i} a_{i} in ω

and

d^{3} = d_{3}, d^{α} = a^{ασ} d_{σ}, d_{α} = a_{ασ} d^{σ} .

Note that the assumption that $(\partial_{1} φ \land \partial_{2} φ) \cdot d > 0$ a.e. in ω is equivalent to the assumption that

d_{3} (y) > 0 for all y \in ω .

(8)

Remark 3. The direction of the vector field d does not depend on its norm. For this reason, it is usually normalized as follows: $| d | = 1$ in Cosserat’s theory of elastic shells, $| d | = 1$ and $d \cdot a_{α} = 0$ (or equivalently $d_{1} = d_{2} = 0$ and $d_{3} = 1$ ) in Kirchhoff-Love theory, $d \cdot a_{3} = 1$ in Relativity theory (where the vector field d is usually called rigging vector field, rather than director field). In order to make this paper useful in all these theories, we study here the general case, where the only assumption on d is that $d \cdot a_{3} > 0$ at all points of ω. □

The next theorem defines a twice-covariant tensor field $k_{αβ} = k_{αβ} (φ, d)$ over ω that will play a crucial role in this paper. It is different from the second fundamental form on the surface $S = φ (ω)$ , but equivalent with it when φ is sufficiently smooth, in the specific way stated below.

Note that the assumption $a_{3} \in W_{loc}^{1, p} (ω)$ (in addition to the assumption $d \in W_{loc}^{1, p} (ω)$ ) is necessary to define the functions $ℓ_{αβ}$ , but it is not necessary to define the functions $k_{αβ}$ .

Note also that formula (9) below generalizes the definition of the covariant derivative of a vector field to the case of immersions with little regularity, i.e., for $φ \in W_{loc}^{1, p} (ω; ℝ^{3})$ with $a_{3} (φ) \in W_{loc}^{1, p} (ω; ℝ^{3})$ , instead of the assumption that $φ \in W_{loc}^{2, p} (ω; ℝ^{3})$ necessary to define the corresponding Christoffel symbols, and that this definition coincide with the classical one when the immersion is of class $W_{loc}^{2, p}$ (see formula (14)).

Theorem 1. Given any vector fields $φ : ω \to ℝ^{3}$ and $d : ω \to ℝ^{3}$ of class $W_{loc}^{1, p}$ , $p > 2$ , such that

(\partial_{1} φ \land \partial_{2} φ) \cdot d > 0 a.e. in ω and a_{3} : = \frac{\partial_{1} φ \land \partial_{2} φ}{| \partial_{1} φ \land \partial_{2} φ |} \in W_{loc}^{1, p} (ω; ℝ^{3}),

let $d_{α} : = d \cdot \partial_{α} φ$ , $d_{3} : = d \cdot a_{3}$ , and

d_{α} |_{β} : = \partial_{α} φ \cdot \partial_{β} (d - d_{3} a_{3}) .

(9)

Then, the functions

k_{αβ} = k_{αβ} (φ, d) : = \frac{1}{2} (\partial_{α} φ \cdot \partial_{β} d + \partial_{α} d \cdot \partial_{β} φ),

(10)

and

ℓ_{αβ} = ℓ_{αβ} (φ, d) : = \frac{1}{d_{3}} (\frac{d_{α} |_{β} + d_{β} |_{α}}{2} - k_{αβ}),

(11)

and

b_{αβ} = b_{αβ} (φ) : = - \partial_{α} φ \cdot \partial_{β} a_{3},

(12)

are well defined, belong to the space $L_{loc}^{p ∕ 2} (ω)$ , and are symmetric in the sense that $k_{αβ} = k_{βα}$ , $ℓ_{αβ} = ℓ_{βα}$ , and $b_{αβ} = b_{βα}$ , a.e. in ω. Besides,

ℓ_{αβ} = b_{αβ},

which means that the functions $ℓ_{αβ}$ defined by (11) are precisely the covariant components of the second fundamental form of the surface $S = φ (ω)$ and thus are independent on the vector field d .

Furthermore, if φ is of class $W_{loc}^{2, p}$ , $p > 2$ , over ω, then the functions $k_{αβ}$ , $ℓ_{αβ}$ and $b_{αβ}$ all belong to the space $L_{loc}^{p} (ω)$ ,

ℓ_{αβ} = b_{αβ} = \partial_{αβ} φ \cdot a_{3} in ω,

(13)

and

d_{α} |_{β} = \partial_{β} d_{α} - Γ_{αβ}^{σ} d_{σ},

(14)

where $Γ_{αβ}^{σ}$ are the Christoffel symbols associated with φ .

Proof. The regularity assumptions on φ and d imply that the functions $k_{αβ}$ and $b_{αβ}$ are well defined, that they belong to the space $L_{loc}^{p ∕ 2} (ω)$ , and that $k_{αβ} = k_{βα}$ . That $b_{αβ} = b_{βα}$ was proved in Ciarlet and Mardare [13, Lemma 3.2].

The assumption that $(\partial_{1} φ \land \partial_{2} φ) \cdot d > 0$ a.e. in ω implies that $d_{3} > 0$ a.e. in ω, and the regularity assumptions on φ and d imply that $d_{α} \in L_{loc}^{p} (ω)$ and $d_{3} \in W_{loc}^{1, p} (ω)$ . Then, $(d_{3} a_{3})$ belongs to $W_{loc}^{1, p} (ω; ℝ^{3})$ since this space is an algebra (remember that $p > 2$ and $ω \subset ℝ^{2}$ ). Then, $ℓ_{αβ}$ is well defined as a measurable function and $ℓ_{αβ} = ℓ_{βα}$ by formula (11).

Let $a_{α} : = \partial_{α} φ$ . Using that $a_{α} \cdot a_{3} = 0$ and $b_{αβ} = b_{βα}$ , we deduce that

\begin{matrix} 2 d_{3} ℓ_{αβ} & = d_{α} |_{β} + d_{β} |_{α} - a_{α} \cdot \partial_{β} d - \partial_{α} d \cdot a_{β} \\ = a_{α} \cdot \partial_{β} (d - d_{3} a_{3}) + a_{β} \cdot \partial_{α} (d - d_{3} a_{3}) - a_{α} \cdot \partial_{β} d - \partial_{α} d \cdot a_{β} \\ = - a_{α} \cdot \partial_{β} (d_{3} a_{3}) - a_{β} \cdot \partial_{α} (d_{3} a_{3}) \\ = - d_{3} (a_{α} \cdot \partial_{β} a_{3} + a_{β} \cdot \partial_{α} a_{3}) \\ = d_{3} b_{αβ} + d_{3} b_{βα} = 2 d_{3} b_{αβ} . \end{matrix}

Hence $ℓ_{αβ} = b_{αβ}$ , as claimed by the theorem. This relation also implies that $ℓ_{αβ} \in L_{loc}^{p ∕ 2} (ω)$ , since we already proved that $b_{αβ} \in L_{loc}^{p ∕ 2} (ω)$ .

Assume now that φ is of class $W_{loc}^{2, p}$ , $p > 2$ , over ω. Then, $a_{α} \in W_{loc}^{1, p} (ω; ℝ^{3})$ , so the functions $k_{αβ}$ and $b_{αβ}$ belong to $L_{loc}^{p} (ω)$ by their definitions (10) and (12). Then, the functions $d_{α} |_{β}$ belong to $L_{loc}^{p} (ω)$ by (9), and finally, $ℓ_{αβ}$ belongs to $L_{loc}^{p} (ω)$ by (11), or by using that $ℓ_{αβ} = b_{αβ} \in L_{loc}^{p} (ω)$ .

Furthermore, $a_{αβ} : = a_{α} \cdot a_{β} \in W_{loc}^{1, p} (ω)$ , which in turn implies that the coefficients $a^{αβ}$ of the inverse matrix field $(a^{αβ}) : = {(a_{αβ})}^{- 1}$ belong to the space $W_{loc}^{1, p} (ω)$ . Then, the vector fields $a^{α}$ defined by the relations $a^{α} \cdot a_{β} = δ_{β}^{α}$ in ω satisfy $a^{α} = a^{αβ} a_{β} \in W_{loc}^{1, p} (ω; ℝ^{3})$ . Besides,

\begin{matrix} d_{α} |_{β} & = a_{α} \cdot \partial_{β} (d - d_{3} a_{3}) \\ = a_{α} \cdot \partial_{β} (d_{σ} a^{σ}) = \partial_{β} d_{α} + d_{σ} (a_{α} \cdot \partial_{β} a^{σ}) \\ = \partial_{β} d_{α} - d_{σ} Γ_{αβ}^{σ} . \end{matrix}

Note that the last equality is a consequence of the Gauss equations (7).

Finally, Schwarz’s theorem about the symmetry of the second derivatives of φ implies that

\begin{matrix} ℓ_{αβ} = b_{αβ} & = - a_{α} \cdot \partial_{β} a_{3} = - \partial_{β} (a_{α} \cdot a_{3}) + \partial_{β} a_{α} \cdot a_{3} \\ = \partial_{βα} φ \cdot a_{3} = \partial_{αβ} φ \cdot a_{3} . \end{matrix}

□

The next theorem is essential. It shows that the coefficients of the moving frame equations associated with a surface with assigned director field depend on the immersion φ only via the functions $a_{αβ}$ , $k_{αβ}$ , $d_{α}$ and $n : = | d |^{2}$ , so these equations generalize the Gauss and Weingarten equation (6) to surfaces with assigned director field. It also paves the way to establishing the counterpart of the classical Gauss–Codazzi equations for surfaces with assigned director field (see Theorems 3 and 5).

Note in passing that the assumption that φ be of class $W_{loc}^{2, p}$ , $p > 2$ , is necessary to define the coefficients $C_{αβ}^{i}$ , while the assumption that φ is only of class $W_{loc}^{1, p}$ , $p > 2$ , suffices to define the coefficients $C_{α 3}^{i}$ .

Theorem 2. Given any surface with assigned director field $(φ, d)$ such that the mapping $φ : ω \to ℝ^{3}$ is of class $W_{loc}^{2, p}$ , $p > 2$ , and the vector field $d : ω \to ℝ^{3}$ is of class $W_{loc}^{1, p}$ , there exist unique functions $C_{αj}^{i} : ω \to ℝ$ of class $L_{loc}^{p}$ such that the vector fields $a_{α} : = \partial_{α} φ$ and d satisfy

\begin{matrix} \partial_{α} a_{β} & = C_{αβ}^{σ} a_{σ} + C_{αβ}^{3} d & in ω, \\ \partial_{α} d & = C_{α 3}^{σ} a_{σ} + C_{α 3}^{3} d & in ω . \end{matrix}

(15)

Besides, the functions $C_{αj}^{i}$ are defined only in terms of the following four tensor fields

\begin{matrix} a_{αβ} = a_{αβ} (φ) & : = \partial_{α} φ \cdot \partial_{β} φ, \\ k_{αβ} = k_{αβ} (φ, d) & : = \frac{1}{2} (\partial_{α} φ \cdot \partial_{β} d + \partial_{β} φ \cdot \partial_{α} d), \\ d_{α} = d_{α} (φ, d) & : = d \cdot \partial_{α} φ, \\ n = n (d) & : = | d |^{2}, \end{matrix}

(16)

by means of the relations

\begin{matrix} C_{αβ}^{3} & = \frac{1}{n - d^{σ} d_{σ}} (\frac{d_{α} |_{β} + d_{β} |_{α}}{2} - k_{αβ}), \\ C_{αβ}^{τ} & = Γ_{αβ}^{τ} + \frac{d^{τ}}{n - d^{σ} d_{σ}} (k_{αβ} - \frac{d_{α} |_{β} + d_{β} |_{α}}{2}), \\ C_{α 3}^{3} & = \frac{d^{β}}{n - d^{σ} d_{σ}} (\frac{d_{α} |_{β} - d_{β} |_{α}}{2} - k_{αβ}) + \frac{\partial_{α} n}{2 (n - d^{σ} d_{σ})}, \\ C_{α 3}^{τ} & = (a^{βτ} + \frac{d^{β} d^{τ}}{n - d^{σ} d_{σ}}) (k_{αβ} - \frac{d_{α} |_{β} - d_{β | α}}{2}) - \frac{d^{τ} \partial_{α} n}{2 (n - d^{σ} d_{σ})}, \end{matrix}

(17)

where

d^{β} : = a^{αβ} d_{α} and d_{α} |_{β} : = \partial_{β} d_{α} - Γ_{αβ}^{σ} d_{σ},

(18)

$(a^{αβ})$ denotes the inverse of the matrix field $(a_{αβ})$ , and $Γ_{αβ}^{τ}$ denote the functions defined in terms of the functions $a_{αβ}$ by

Γ_{αβ}^{τ} : = \frac{1}{2} a^{στ} (\partial_{α} a_{βσ} + \partial_{β} a_{ασ} - \partial_{σ} a_{αβ}) .

(19)

Note that the denominators in (17) do not vanish since the assumption that $(\partial_{1} φ \land \partial_{2} φ) \cdot d > 0$ in ω implies that

n - d^{σ} d_{σ} = \frac{{((\partial_{1} φ \land \partial_{2} φ) \cdot d)}^{2}}{| \partial_{1} φ \land \partial_{2} φ |^{2}} > 0 in ω .

Remark 4. If $d = a_{3} : = (\partial_{1} φ \land \partial_{2} φ) ∕ | \partial_{1} φ \land \partial_{2} φ |$ , then $d_{α} = 0$ and $n = 1$ and relations (17) become

\begin{matrix} C_{αβ}^{3} = - k_{αβ} = b_{αβ}, C_{αβ}^{τ} = Γ_{αβ}^{τ}, C_{α 3}^{3} = 0, C_{α 3}^{τ} = - b_{α}^{τ} . \end{matrix}

This proves that system (15) reduces in this case to the classical Gauss and Weingarten equations satisfied by the surface $S = φ (ω)$ . □

Proof of Theorem 2. The assumption that $(φ, d)$ is a surface with assigned director field (see Definition 1) implies that $(\partial_{1} φ \land \partial_{2} φ) \cdot d > 0$ in ω. Then, the vectors fields ${a_{1}, a_{2}, d}$ form a basis in $ℝ^{3}$ at every point of ω, so the functions $C_{αj}^{i}$ are nothing but the coefficients of the vector fields $\partial_{α} a_{β}$ and $\partial_{α} d$ over this basis.

Next observe that the moving frame equation (15) hold if and only if

\begin{matrix} C_{αβ}^{σ} a_{στ} + C_{αβ}^{3} d_{τ} & = \partial_{α} a_{β} \cdot a_{τ}, \\ C_{αβ}^{σ} d_{σ} + C_{αβ}^{3} d_{i} d^{i} & = \partial_{α} a_{β} \cdot d, \\ C_{α 3}^{σ} a_{στ} + C_{α 3}^{3} d_{τ} & = \partial_{α} d \cdot a_{τ}, \\ C_{α 3}^{σ} d_{σ} + C_{α 3}^{3} d_{i} d^{i} & = \partial_{α} d \cdot d . \end{matrix}

Besides, the definition of the functions $k_{αβ}$ implies that

\begin{matrix} k_{αβ} & = \frac{1}{2} (\partial_{β} (\partial_{α} φ \cdot d) + \partial_{α} (\partial_{β} φ \cdot d)) - \partial_{αβ} φ \cdot d \\ = \frac{\partial_{α} d_{β} + \partial_{β} d_{α}}{2} - \partial_{αβ} φ \cdot d, \end{matrix}

which in turn implies that

\partial_{αβ} φ \cdot d = \frac{\partial_{α} d_{β} + \partial_{β} d_{α}}{2} - k_{αβ} .

Using these relations and the definition (19) of the Christoffel symbols $Γ_{αβ}^{σ}$ , one deduces that the above system is equivalent to the system

\begin{matrix} C_{αβ}^{σ} + C_{αβ}^{3} d^{σ} & = Γ_{αβ}^{σ}, \\ C_{αβ}^{σ} d_{σ} + C_{αβ}^{3} d_{i} d^{i} & = \frac{\partial_{α} d_{β} + \partial_{β} d_{α}}{2} - k_{αβ}, \\ C_{α 3}^{σ} + C_{α 3}^{3} d^{σ} & = (\partial_{α} d_{τ} - \frac{\partial_{α} d_{τ} + \partial_{τ} d_{α}}{2} + k_{ατ}) a^{στ}, \\ C_{α 3}^{σ} d_{σ} + C_{α 3}^{3} d_{i} d^{i} & = \frac{1}{2} \partial_{α} (d_{i} d^{i}), \end{matrix}

which in turn is equivalent to the system

\begin{matrix} C_{αβ}^{σ} & = Γ_{αβ}^{σ} - C_{αβ}^{3} d^{σ}, \\ {(d_{3})}^{2} C_{αβ}^{3} & = \frac{\partial_{α} d_{β} + \partial_{β} d_{α}}{2} - k_{αβ} - Γ_{αβ}^{σ} d_{σ}, \\ C_{α 3}^{σ} & = a^{στ} (\frac{\partial_{α} d_{τ} - \partial_{τ} d_{α}}{2} + k_{ατ}) - C_{α 3}^{3} d^{σ}, \\ {(d_{3})}^{2} C_{α 3}^{3} & = \frac{1}{2} \partial_{α} (d_{i} d^{i}) - d^{τ} (\frac{\partial_{α} d_{τ} - \partial_{τ} d_{α}}{2} + k_{ατ}) . \end{matrix}

Since $d_{α} |_{β} = \partial_{β} d_{α} - Γ_{αβ}^{σ} d_{σ}$ , $n = d \cdot d = d^{i} d_{i}$ , and ${(d_{3})}^{2} = n - d^{σ} d_{σ} > 0$ by the definition of a surface with assigned director field, the above system is equivalent to system (17).

This completes the proof of the theorem. □

The next three corollaries are consequences of Theorems 1 and 2. The first one shows that system (17) can be recast in a simpler form by replacing the functions $k_{αβ}$ appearing in its right-hand side by the functions $ℓ_{αβ}$ introduced in Theorem 1. The statement and proof of this corollary do not make any reference to any mapping φ and vector field d that might possibly be the source of the given functions $a_{αβ}$ , $k_{αβ}$ , $d_{α}$ , and n, in view of its application in the intrinsic theory of nonlinearly elastic shells.

The second corollary establishes the moving frame equations for the vector fields ${a^{1}, a^{2}, d}$ , which are the counterpart for surfaces with assigned director field of the system (7) for surfaces.

The third corollary gives a necessary and sufficient condition for the regularity of a surface with assigned director field.

Corollary 1. Let ω be an open subset of $ℝ^{2}$ and let $p \in (2, + \infty]$ . Given any symmetric and positive-definite matrix field $(a_{αβ}) \in W_{loc}^{1, p} (ω; ℝ^{2 \times 2})$ , any symmetric matrix $(k_{αβ}) \in L_{loc}^{p} (ω; ℝ^{2 \times 2})$ , any vector field $(d_{α}) \in W_{loc}^{1, p} (ω; ℝ^{2})$ , and any scalar field $n \in W_{loc}^{1, p} (ω)$ such that

(n - a^{αβ} d_{α} d_{β}) (y) > 0 for all y \in ω,

where $(a^{αβ})$ denotes the inverse of the matrix field $(a_{αβ})$ , let $d_{3} : = {(n - a^{αβ} d_{α} d_{β})}^{1 ∕ 2}$ , let

\begin{matrix} Γ_{αβ}^{τ} & : = \frac{1}{2} a^{στ} (\partial_{α} a_{βσ} + \partial_{β} a_{ασ} - \partial_{σ} a_{αβ}), \\ d^{α} & : = a^{αβ} d_{β}, \end{matrix}

let

d_{α} |_{β} : = \partial_{α} d_{β} - Γ_{αβ}^{σ} d_{σ} and d^{α} |_{β} : = \partial_{β} d^{α} + Γ_{βσ}^{α} d^{σ},

and let

ℓ_{αβ} : = \frac{1}{d_{3}} (\frac{d_{α} |_{β} + d_{β} |_{α}}{2} - k_{αβ}) and ℓ_{α}^{τ} : = a^{τσ} ℓ_{ασ} .

Then, the following relations (which are identical to relations (17) in Theorem 2)

\begin{matrix} C_{αβ}^{3} & = \frac{1}{n - d^{σ} d_{σ}} (\frac{d_{α} |_{β} + d_{β} |_{α}}{2} - k_{αβ}), \\ C_{αβ}^{τ} & = Γ_{αβ}^{τ} + \frac{d^{τ}}{n - d^{σ} d_{σ}} (k_{αβ} - \frac{d_{α} |_{β} + d_{β} |_{α}}{2}), \\ C_{α 3}^{3} & = \frac{d^{β}}{n - d^{σ} d_{σ}} (\frac{d_{α} |_{β} - d_{β} |_{α}}{2} - k_{αβ}) + \frac{\partial_{α} n}{2 (n - d^{σ} d_{σ})}, \\ C_{α 3}^{τ} & = (a^{βτ} + \frac{d^{β} d^{τ}}{n - d^{σ} d_{σ}}) (k_{αβ} - \frac{d_{α} |_{β} - d_{β | α}}{2}) - \frac{d^{τ} \partial_{α} n}{2 (n - d^{σ} d_{σ})}, \end{matrix}

(20)

are equivalent to the relations

\begin{matrix} C_{αβ}^{3} & = \frac{1}{d_{3}} ℓ_{αβ}, \\ C_{αβ}^{τ} & = Γ_{αβ}^{τ} - \frac{d^{τ}}{d_{3}} ℓ_{αβ}, \\ C_{α 3}^{3} & = \frac{1}{d_{3}} (\partial_{α} d_{3} + d^{β} ℓ_{αβ}), \\ C_{α 3}^{τ} & = - d_{3} ℓ_{α}^{τ} + d^{τ} |_{α} - \frac{d^{τ}}{d_{3}} (\partial_{α} d_{3} + d^{β} ℓ_{αβ}) . \end{matrix}

(21)

Proof. The definition of the functions $d_{3}$ and $ℓ_{αβ}$ implies that

\frac{1}{n - d^{σ} d_{σ}} (\frac{d_{α} |_{β} + d_{β} |_{α}}{2} - k_{αβ}) = \frac{1}{d_{3}} ℓ_{αβ},

and

Γ_{αβ}^{τ} + \frac{d^{τ}}{n - d^{σ} d_{σ}} (k_{αβ} - \frac{d_{α} |_{β} + d_{β} |_{α}}{2}) = Γ_{αβ}^{τ} - \frac{d^{τ}}{d_{3}} ℓ_{αβ} .

This proves the equivalence of the first and of the second equations of systems (20) and (21).

The definition of the Christoffel symbols $Γ_{αβ}^{τ}$ implies that the covariant derivatives of $a^{βτ}$ vanish, i.e.,

a^{βτ} |_{α} : = \partial_{α} a^{βτ} + Γ_{ασ}^{β} a^{στ} + Γ_{ασ}^{τ} a^{βσ} = 0 in ω .

Then, the definition of the covariant derivatives $d_{α} |_{β}$ implies that

\begin{matrix} \partial_{α} (a^{σβ} d_{σ} d_{β}) & = \partial_{α} a^{σβ} d_{σ} d_{β} + a^{σβ} \partial_{α} d_{σ} d_{β} + a^{σβ} d_{σ} \partial_{α} d_{β} \\ = (\partial_{α} a^{σβ} d_{σ} d_{β} + a^{σβ} Γ_{ασ}^{τ} d_{τ} d_{β} + a^{σβ} d_{σ} Γ_{αβ}^{τ} d_{τ}) \\ + a^{σβ} d_{σ} |_{α} d_{β} + a^{σβ} d_{σ} d_{β} |_{α} \\ = (\partial_{α} a^{σβ} + Γ_{ατ}^{σ} a^{τβ} + Γ_{ατ}^{β} a^{στ}) d_{σ} d_{β} + a^{σβ} d_{σ} |_{α} d_{β} + a^{σβ} d_{σ} d_{β} |_{α} \\ = a^{σβ} d_{σ} |_{α} d_{β} + a^{σβ} d_{σ} d_{β} |_{α}, \end{matrix}

so that

\begin{matrix} \frac{\partial_{α} n}{2 (n - d^{σ} d_{σ})} & = \frac{\partial_{α} ({(d_{3})}^{2} + a^{σβ} d_{σ} d_{β})}{2 {(d_{3})}^{2}} \\ = \frac{\partial_{α} d_{3}}{d_{3}} + \frac{a^{σβ} d_{σ} |_{α} d_{β} + a^{σβ} d_{σ} d_{β} |_{α}}{2 {(d_{3})}^{2}} \\ = \frac{\partial_{α} d_{3}}{d_{3}} + \frac{a^{σβ} d_{σ} d_{β} |_{α}}{{(d_{3})}^{2}} . \end{matrix}

Next, we deduce by using again the definition of the functions $d_{3}$ and $ℓ_{αβ}$ that, on the one hand,

\begin{matrix} \frac{d^{β}}{n - d^{σ} d_{σ}} (\frac{d_{α} |_{β} - d_{β} |_{α}}{2} - k_{αβ}) + \frac{\partial_{α} n}{2 (n - d^{σ} d_{σ})} \\ = \frac{d^{β}}{{(d_{3})}^{2}} (d_{3} ℓ_{αβ} - d_{β} |_{α}) + \frac{\partial_{α} d_{3}}{d_{3}} + \frac{d^{β} d_{β} |_{α}}{{(d_{3})}^{2}} \\ = \frac{1}{d_{3}} (\partial_{α} d_{3} + d^{β} ℓ_{αβ}), \end{matrix}

and, on the other hand, that

\begin{matrix} (a^{βτ} + \frac{d^{β} d^{τ}}{n - d^{σ} d_{σ}}) (k_{αβ} - \frac{d_{α} |_{β} - d_{β | α}}{2}) - \frac{d^{τ} \partial_{α} n}{2 (n - d^{σ} d_{σ})} \\ = (a^{βτ} + \frac{d^{β} d^{τ}}{{(d_{3})}^{2}}) (d_{β} |_{α} - d_{3} ℓ_{αβ}) - d^{τ} (\frac{\partial_{α} d_{3}}{d_{3}} + \frac{a^{σβ} d_{σ} d_{β} |_{α}}{{(d_{3})}^{2}}) \\ = a^{βτ} (d_{β} |_{α} - d_{3} ℓ_{αβ}) - \frac{d^{β} d^{τ}}{{(d_{3})}^{2}} d_{3} ℓ_{αβ} - d^{τ} \frac{\partial_{α} d_{3}}{d_{3}} \\ = d^{τ} |_{α} - d_{3} ℓ_{α}^{τ} - \frac{d^{τ}}{d_{3}} (\partial_{α} d_{3} + d^{β} ℓ_{αβ}) . \end{matrix}

This proves the equivalence of the third and of the fourth equations of the systems (20) and (21). □

Corollary 2. Given any surface with assigned director field $(φ, d)$ such that the mapping $φ : ω \to ℝ^{3}$ is of class $W_{loc}^{2, p}$ , $p > 2$ , and the vector field $d : ω \to ℝ^{3}$ is of class $W_{loc}^{1, p}$ , the three vector fields ${a^{1}, a^{2}, d}$ , where

a^{τ} : = a^{τσ} a_{σ}, a_{σ} : = \partial_{σ} φ, a_{αβ} : = a_{α} \cdot a_{β}, (a^{τσ}) : = {(a_{αβ})}^{- 1},

satisfy the system

\begin{matrix} \partial_{α} a^{τ} & = (a^{βτ} a_{σλ} (C_{αβ}^{σ} - Γ_{αβ}^{σ}) - Γ_{αλ}^{τ}) a^{λ} + (a^{βτ} C_{αβ}^{3}) d, \\ \partial_{α} d & = (a_{σλ} C_{α 3}^{σ}) a^{λ} + C_{α 3}^{3} d . \end{matrix}

Proof. Using the definition of the vectors fields $a^{τ}$ , we have

\partial_{α} a^{τ} = \partial_{α} (a^{βτ} a_{β}) = a^{βτ} \partial_{α} a_{β} + (\partial_{α} a^{βτ}) a_{β},

Since the covariant derivatives of the contravariant components of the metric tensor vanish in ω, i.e.,

a^{βτ} |_{α} : = \partial_{α} a^{βτ} + Γ_{ασ}^{β} a^{στ} + Γ_{ασ}^{τ} a^{βσ} = 0,

we next have

\partial_{α} a^{τ} = a^{βτ} \partial_{α} a_{β} - (Γ_{ασ}^{β} a^{στ} + Γ_{ασ}^{τ} a^{βσ}) a_{β} .

Then, using the first equation of system (15), i.e., $\partial_{α} a_{β} = C_{αβ}^{σ} a_{σ} + C_{αβ}^{3} d$ , in the right-hand side of the above relation, then using the relations $a_{σ} = a_{σλ} a^{λ}$ , we finally have

\begin{matrix} \partial_{α} a^{τ} & = a^{βτ} (C_{αβ}^{σ} a_{σ} + C_{αβ}^{3} d) - (Γ_{ασ}^{β} a^{στ} + Γ_{ασ}^{τ} a^{βσ}) a_{β} \\ = a^{βτ} C_{αβ}^{σ} (a_{σλ} a^{λ}) - (Γ_{ασ}^{β} a^{στ} + Γ_{ασ}^{τ} a^{βσ}) (a_{βλ} a^{λ}) + a^{βτ} C_{αβ}^{3} d \\ = a^{βτ} C_{αβ}^{σ} a_{σλ} a^{λ} - Γ_{ασ}^{β} a^{στ} a_{βλ} a^{λ} - Γ_{ασ}^{τ} a^{βσ} a_{βλ} a^{λ} + a^{βτ} C_{αβ}^{3} d \\ = a^{βτ} C_{αβ}^{σ} a_{σλ} a^{λ} - Γ_{αβ}^{σ} a^{βτ} a_{σλ} a^{λ} - Γ_{αλ}^{τ} a^{λ} + a^{βτ} C_{αβ}^{3} d \\ = (a^{βτ} (C_{αβ}^{σ} - Γ_{αβ}^{σ}) a_{σλ} - Γ_{αλ}^{τ}) a^{λ} + (a^{βτ} C_{αβ}^{3}) d . \end{matrix}

Besides, using that $a_{σ} = a_{σλ} a^{λ}$ in the right-hand side of the second equation of system (15) yields the equation

\begin{matrix} \partial_{α} d & = C_{α 3}^{σ} a_{σ} + C_{α 3}^{3} d, \\ = (C_{α 3}^{σ} a_{σλ}) a^{α} + C_{α 3}^{3} d . \end{matrix}

This completes the proof of the corollary. □

Corollary 3. Let $p \in (2, + \infty]$ and an integer $k \geq 1$ . A surface with assigned director field $(φ, d)$ such that the mapping $φ : ω \to ℝ^{3}$ is of class $W_{loc}^{2, p} (ω; ℝ^{3})$ and the vector field $d : ω \to ℝ^{3}$ is of class $W_{loc}^{1, p} (ω; ℝ^{3})$ , satisfies

φ \in W_{loc}^{k + 1, p} (ω; ℝ^{3}) and d \in W_{loc}^{k, p} (ω; ℝ^{3})

if and only if the functions defined by (16) satisfy

a_{αβ} \in W_{loc}^{k, p} (ω), k_{αβ} \in W_{loc}^{k - 1, p} (ω), d_{α} \in W_{loc}^{k, p} (ω), and n \in W_{loc}^{k, p} (ω) .

Proof. If $φ \in W_{loc}^{k + 1, p} (ω; ℝ^{3})$ and $d \in W_{loc}^{k, p} (ω; ℝ^{3})$ , then we infer from relations (16) that the functions $a_{αβ}$ , $d_{α}$ and n belong to $W_{loc}^{k, p} (ω)$ and that $k_{αβ} \in W_{loc}^{k - 1, p} (ω)$ .

If the functions $a_{αβ}$ , $d_{α}$ , and n belong to $W_{loc}^{k, p} (ω)$ and $k_{αβ}$ belongs to $W_{loc}^{k - 1, p} (ω)$ , then the functions $a^{αβ}$ and $(n - a^{στ} d_{σ} d_{τ})$ belong to the space $W_{loc}^{k, p} (ω)$ , so they are continuous since $W_{loc}^{k, p} (ω) \subset C^{k - 1} (ω)$ by the Sobolev’s embedding theorem, and they satisfy

n - a^{στ} d_{σ} d_{τ} > 0 at every point of ω .

Note that this inequality is a consequence of the assumption of the theorem that $(φ, d)$ is a surface with assigned director field, which means that

(\partial_{1} φ \land \partial_{2} φ) \cdot d > 0 at every point of ω .

Indeed, this last inequality implies the two vector fields ${\partial_{1} φ, \partial_{2} φ}$ are linearly independent at every point of ω, so the vector field $a_{3} : = (\partial_{1} φ \land \partial_{2} φ) ∕ | \partial_{1} φ \land \partial_{2} φ |$ is well defined and the matrix field $(a_{αβ})$ is symmetric and positive-definite at every point of ω. Then, the inverse matrix field $(a^{αβ}) : = {(a_{αβ})}^{- 1}$ is also symmetric and positive-definite at every point of ω and

\begin{matrix} n - a^{στ} d_{σ} d_{τ} & = (d - d_{σ} a^{σα} \partial_{α} φ) \cdot (d - d_{τ} a^{τβ} \partial_{β} φ) \\ = | (d \cdot a_{3}) a_{3} |^{2} = {(d \cdot a_{3})}^{2} \\ = \frac{{((\partial_{1} φ \land \partial_{2} φ) \cdot d)}^{2}}{| \partial_{1} φ \land \partial_{2} φ |^{2}} > 0 in ω . \end{matrix}

It follows from the above properties that the function ${(n - a^{στ} d_{σ} d_{τ})}^{- 1}$ belongs to the space $W_{loc}^{k, p} (ω)$ and that the functions $Γ_{αβ}^{τ}$ belong to the space $W_{loc}^{k - 1, p} (ω)$ . Then, we infer from relations (17) and from the fact that the space $W_{loc}^{k, p} (ω)$ is an algebra that the functions $C_{αj}^{i}$ belong to the space $W_{loc}^{k - 1, p} (ω)$ . Then, system (15), which is satisfied with these coefficients $C_{αj}^{i}$ thanks to Theorem 2, implies that the vector fields $\partial_{α} a_{β}$ and $\partial_{α} d$ belong to the space $W_{loc}^{k - 1, p} (ω)$ , which in turn imply that $a_{β} \in W_{loc}^{k, p} (ω; ℝ^{3})$ and $d \in W_{loc}^{k, p} (ω; ℝ^{3})$ . Finally, $φ \in W_{loc}^{k + 1, p} (ω; ℝ^{3})$ since $\partial_{β} φ = a_{β}$ . □

We are now in a position to prove that the functions $a_{αβ}$ , $k_{αβ}$ , $d_{α}$ , and n associated with a surface with director field $(φ, d)$ necessarily satisfy compatibility conditions (see relation (22) in Theorem 3 below) that can be seen as the counterpart in the theory of surfaces with assigned director field of the classical Gauss–Codazzi equations in surface theory (see Theorem 5).

Theorem 3. Given any surface with assigned director field $(φ, d)$ such that the mapping $φ : ω \to ℝ^{3}$ is of class $W_{loc}^{2, p}$ , $p > 2$ , and the vector field $d : ω \to ℝ^{3}$ is of class $W_{loc}^{1, p}$ , let $a_{αβ}$ , $k_{αβ}$ , $d_{α}$ , and n, be the functions defined by (16).

Then, the functions $C_{αj}^{i}$ defined in terms of $(a_{αβ}, k_{αβ}, d_{α}, n)$ by relations (17) are of class $L_{loc}^{p}$ and satisfy the compatibility conditions

\partial_{α} C_{β} + C_{α} C_{β} = \partial_{β} C_{α} + C_{β} C_{α} in D^{'} (ω; ℝ^{3 \times 3}),

(22)

where $C_{α} : = (C_{αj}^{i})$ denotes the matrix field with the Christoffel symbol $C_{αj}^{i}$ at its i -th row and j -th column.

Proof. Let $F : = (\begin{matrix} \partial_{1} φ | \partial_{2} φ | d \end{matrix})$ denote the matrix field with $\partial_{1} φ$ , $\partial_{2} φ$ , and d as its column vector fields. Then, system (15) is equivalent to the matrix equation

\partial_{α} F = F C_{α} in ω .

This matrix equation being a Pfaff system of the form studied in S. Mardare [14], its coefficients must satisfy the compatibility conditions

\partial_{α} C_{β} + C_{α} C_{β} = \partial_{β} C_{α} + C_{β} C_{α} in D^{'} (ω; ℝ^{3 \times 3}),

as proven in S. Mardare [14].

This completes the proof of the theorem. □

4. Recovery of a surface with assigned director field

In this section, we show that a simply connected surface with assigned director field can be defined in an intrinsic manner, up to a proper isometry of $ℝ^{3}$ , by specifying the tensor fields $a_{αβ}$ , $k_{αβ}$ , $d_{α}$ , and n. We recall that a proper isometry of $ℝ^{3}$ is a mapping $r : ℝ^{3} \to ℝ^{3}$ of the form

r (x) = a + B x for all x \in ℝ^{3},

where a is a vector in $ℝ^{3}$ and B is a special orthogonal matrix of order three.

The next theorem generalizes the fundamental theorem of surface theory (see, e.g., [4, 6, 7, 15 –18]) to surfaces with assigned director field. The classical theorem is obtained by letting $d_{α} = 0$ and $n = 1$ in Theorem 4. Remember that the space $W_{loc}^{1, p} (ω)$ , $p > 2$ , is identified with its image in $C^{0} (ω)$ by the Sobolev embedding theorem.

Theorem 4. Let ω be a simply connected open subset of $ℝ^{2}$ and $p \in (2, + \infty]$ . Given any positive-definite symmetric matrix $(a_{αβ}) : ω \to ℝ^{2 \times 2}$ of class $W_{loc}^{1, p}$ , any symmetric matrix $(k_{αβ}) : ω \to ℝ^{2 \times 2}$ of class $L_{loc}^{p}$ , any vector field $(d_{α}) : ω \to ℝ^{2}$ of class $W_{loc}^{1, p}$ , and any scalar field $n : ω \to ℝ$ of class $W_{loc}^{1, p}$ satisfying

(n - a^{αβ} d_{α} d_{β}) (y) > 0 for all y \in ω,

where $a^{αβ}$ denotes the components of matrix field $(a^{αβ}) : = {(a_{αβ})}^{- 1}$ , let $C_{αj}^{i} : ω \to ℝ$ denote the functions defined by relations (17).

Assume that the matrix fields $C_{α} : = (C_{αj}^{i})$ satisfy the compatibility conditions (22). Then, there exists a surface with assigned director field $(φ, d)$ with $φ : ω \to ℝ^{3}$ of class $W_{loc}^{2, p}$ and $d : ω \to ℝ^{3}$ of class $W_{loc}^{1, p}$ such that

\begin{matrix} a_{αβ} & = \partial_{α} φ \cdot \partial_{β} φ, \\ k_{αβ} & = \frac{1}{2} (\partial_{α} φ \cdot \partial_{β} d + \partial_{α} d \cdot \partial_{β} φ), \\ d_{α} & = \partial_{α} φ \cdot d, \\ n & = | d |^{2} . \end{matrix}

(23)

Besides, if another surface with assigned director field $(\tilde{φ}, \tilde{d})$ with $\tilde{φ} : ω \to ℝ^{3}$ of class $W_{loc}^{2, p}$ and $\tilde{d} : ω \to ℝ^{3}$ of class $W_{loc}^{1, p}$ satisfy the above system, then there exists a vector $a \in ℝ^{3}$ and a special orthogonal matrix $R \in ℝ^{3 \times 3}$ such that

\begin{matrix} \tilde{φ} (y) & = a + R φ (y) & in ω, \\ \tilde{d} (y) & = R d (y) & in ω . \end{matrix}

Proof. Let there be given a point $y_{0}$ in ω and a matrix $F_{0}$ in $ℝ^{3 \times 3}$ . We will choose later in the proof a specific matrix $F_{0}$ , but for the time being this choice is irrelevant.

Since the matrix fields $C_{α}$ are of class $L_{loc}^{p}$ and satisfy the compatibility conditions (22), there exist a (unique) matrix field $F = (F_{j}^{i}) : ω \to ℝ^{3 \times 3}$ of class $W_{loc}^{1, p}$ satisfying the Pfaff system

\begin{matrix} \partial_{α} F & = F C_{α} in ω, \\ F (y_{0}) & = F_{0}, \end{matrix}

(24)

as proven in S. Mardare [14]. Then, the column vector fields of F , henceforth denoted

a_{1} : = {(F_{1}^{i})}_{i = 1}^{3}, a_{2} : = {(F_{2}^{i})}_{i = 1}^{3}, d : = {(F_{3}^{i})}_{i = 1}^{3},

satisfy the equations

\begin{matrix} \partial_{α} a_{β} & = C_{αβ}^{σ} a_{σ} + C_{αβ}^{3} d & in ω, \\ \partial_{α} d & = C_{α 3}^{σ} a_{σ} + C_{α 3}^{3} d & in ω . \end{matrix}

(25)

Since the matrix fields $(a_{αβ})$ and $(k_{αβ})$ are symmetric by assumption, an inspection of the definition (17) of the functions $C_{αj}^{i}$ shows that

C_{αβ}^{σ} = C_{βα}^{σ} and C_{αβ}^{3} = C_{βα}^{3} .

Then, the first equation of system (25) implies that

\partial_{α} a_{β} = \partial_{β} a_{α} in ω,

which next implies that there exists a mapping $φ : ω \to ℝ^{3}$ of class $W_{loc}^{2, p}$ that satisfies the Poincaré system:

\partial_{α} φ = a_{α} in ω .

(26)

This existence theorem is classical (see, e.g., Ciarlet [19]) but also a particular case of the theorem of S. Mardare [14] already used above to prove the existence of a solution to the Pfaff system (24). Note that the uniqueness of such a solution can be ensured by asking the solution to satisfy in addition $φ (y_{0}) = φ_{0}$ , where $φ_{0}$ is any given vector in $ℝ^{3}$ , but this is not useful for our purpose here.

It remains to prove that the vector field $d : ω \to ℝ^{3}$ and the mapping $φ : ω \to ℝ^{3}$ defined above satisfy the equations of system (23) and that the pair $(φ, d)$ is a surface with assigned director field according to Definition 1. We will prove that this is indeed the case provided we choose the matrix $F_{0}$ appropriately. To this end, we first prove that there exist three vectors in $ℝ^{3}$ , henceforth denoted as $a_{1}^{0}$ , $a_{2}^{0}$ , and $d^{0}$ , such that

\begin{matrix} a_{α}^{0} \cdot a_{β}^{0} & = a_{αβ} (y_{0}), \\ a_{α}^{0} \cdot d^{0} & = d_{α} (y_{0}), \\ | d^{0} |^{2} & = n (y_{0}), \end{matrix}

(27)

and

d^{0} \cdot a_{3}^{0} > 0, where a_{3}^{0} : = \frac{a_{1}^{0} \land a_{2}^{0}}{| a_{1}^{0} \land a_{2} |},

then we define

F_{0} : = (\begin{matrix} a_{1}^{0} | a_{2}^{0} | d^{0} \end{matrix})

as the matrix whose column vectors are $a_{1}^{0}$ , $a_{2}^{0}$ , and $d^{0}$ , in this order.

That vectors $a_{1}^{0}$ , $a_{2}^{0}$ , and $d^{0}$ satisfying the above system do exist is proven as follows. Since the matrix $(\begin{matrix} a_{αβ} (y_{0}) \end{matrix})$ isa positive-definite symmetric matrix of order two, there exists a unique symmetric and positive-definite matrix $(\begin{matrix} r_{αβ} \end{matrix})$ of order two such that

{(\begin{matrix} r_{αβ} \end{matrix})}^{2} = (\begin{matrix} a_{αβ} (y_{0}) \end{matrix}) .

Then, the two vectors

a_{α}^{0} : = (\begin{matrix} r_{α 1}, r_{α 2}, 0 \end{matrix}) \in ℝ^{3}

are linearly independent, since the $3 \times 2$ matrix with $a_{1}^{0}$ and $a_{2}^{0}$ as its two column vectors has rank two, the same as the rank of the positive-definite symmetric matrix $(\begin{matrix} r_{αβ} \end{matrix})$ . Hence, the vector

a_{3}^{0} : = \frac{a_{1}^{0} \land a_{2}^{0}}{| a_{1}^{0} \land a_{2}^{0} |} \in ℝ^{3}

is well defined, and in fact, $a_{3}^{0} = (0, 0, 1)$ , which in turn implies that the vector

d^{0} : = a^{αβ} (y_{0}) d_{α} (y_{0}) a_{β}^{0} + {(n (y_{0}) - a^{αβ} (y_{0}) d_{α} (y_{0}) d_{β} (y_{0}))}^{1 ∕ 2} a_{3}^{0} \in ℝ^{3}

is also well defined.

The above definitions of the vectors $a_{α}^{0}$ and $d^{0}$ imply that they satisfy the relations

\begin{matrix} a_{α}^{0} \cdot a_{β}^{0} & = a_{αβ} (y_{0}), \\ a_{α}^{0} \cdot d^{0} & = d_{α} (y_{0}), \\ | d^{0} |^{2} & = n (y_{0}), \\ d^{0} \cdot a_{3}^{0} & > 0, \end{matrix}

as desired.

We are now in a position to prove that the vector field $d : ω \to ℝ^{3}$ and the mapping $φ : ω \to ℝ^{3}$ defined at the beginning of this proof satisfy all the equations of system (23).

First, since the matrix field F satisfies $F (y_{0}) = F_{0}$ (see the second equation of system (24)), their column vector fields satisfy

a_{α} (y_{0}) = a_{α}^{0} and d (y_{0}) = d^{0} .

Then, we infer from the previous system that

\begin{matrix} (a_{α} \cdot a_{β}) (y_{0}) & = a_{αβ} (y_{0}), \\ (a_{α} \cdot d) (y_{0}) & = d_{α} (y_{0}), \\ | d (y_{0}) |^{2} & = n (y_{0}), \\ (d \cdot a_{3}) (y_{0}) & > 0 . \end{matrix}

(28)

Second, we infer from system (25) that the functions $(a_{α} \cdot a_{β})$ , $(a_{α} \cdot d)$ , and $| d |^{2}$ defined from ω into ℝ satisfy the system

\begin{matrix} \partial_{σ} (a_{α} \cdot a_{β}) & = C_{σα}^{τ} (a_{τ} \cdot a_{β}) + C_{σα}^{3} (d \cdot a_{β}) + C_{σβ}^{τ} (a_{τ} \cdot a_{α}) + C_{σβ}^{3} (d \cdot a_{α}), \\ \partial_{σ} (a_{α} \cdot d) & = C_{σα}^{τ} (a_{τ} \cdot d) + C_{σα}^{3} (d \cdot d) + C_{σ 3}^{τ} (a_{τ} \cdot a_{α}) + C_{σ 3}^{3} (d \cdot a_{α}), \\ \partial_{σ} (| d |^{2}) & = 2 C_{σ 3}^{τ} (a_{τ} \cdot d) + 2 C_{σ 3}^{3} (d \cdot d) . \end{matrix}

(29)

Third, we infer from the definition of the functions $C_{αj}^{i}$ (see (17) and (18)) that the functions $a_{αβ}$ , $d_{α}$ , and n defined from ω into ℝ satisfy the system

\begin{matrix} C_{σα}^{τ} a_{τβ} + C_{σα}^{3} d_{β} + C_{σβ}^{τ} a_{τα} + C_{σβ}^{3} d_{α} = \partial_{σ} a_{αβ}, \\ C_{σα}^{τ} d_{τ} + C_{σα}^{3} n + C_{σ 3}^{τ} a_{τα} + C_{σ 3}^{3} d_{α} = \partial_{σ} d_{α}, \\ 2 C_{σ 3}^{τ} d_{τ} + 2 C_{σ 3}^{3} n = \partial_{σ} n . \end{matrix}

(30)

The last three systems (28)–(30) together show that

\begin{matrix} a_{α} \cdot a_{β} & = a_{αβ} & in ω, \\ a_{α} \cdot d & = d_{α} & in ω, \\ | d |^{2} & = n & in ω, \end{matrix}

(31)

since the functions appearing in the left-hand sides of (31) satisfy the Pfaff system (29), while the functions appearing in the right-hand sides of (31) satisfy the Pfaff system (30), which have the same coefficients as system (29). Since the equalities in system (31) hold at the point $y_{0} \in ω$ according to relations (28), they hold at all points of ω by the existence and uniqueness result for Pfaff systems proved in S. Mardare [20].

It remains to establish the second equation of (23), viz., that

\frac{1}{2} (\partial_{α} φ \cdot \partial_{β} d + \partial_{α} d \cdot \partial_{β} φ) = k_{αβ} in ω,

or equivalently that

\frac{1}{2} (a_{α} \cdot \partial_{β} d + \partial_{α} d \cdot a_{β}) = k_{αβ} in ω,

since $\partial_{α} φ = a_{α}$ by (26).

The definition (17) of the functions $C_{α 3}^{τ}$ and $C_{α 3}^{3}$ , viz.,

C_{α 3}^{3} = \frac{d^{β}}{n - d^{σ} d_{σ}} (\frac{d_{α} |_{β} - d_{β} |_{α}}{2} - k_{αβ}) + \frac{\partial_{α} n}{2 (n - d^{σ} d_{σ})}

and

C_{α 3}^{τ} = (a^{βτ} + \frac{d^{β} d^{τ}}{n - d^{σ} d_{σ}}) (k_{αβ} - \frac{d_{α} |_{β} - d_{β | α}}{2}) - \frac{d^{τ} \partial_{α} n}{2 (n - d^{σ} d_{σ})},

together imply that

C_{α 3}^{τ} = a^{βτ} (k_{αβ} - \frac{d_{α | β} - d_{β | α}}{2}) - d^{τ} C_{α 3}^{3},

(32)

on the one hand. On the other hand, the second equation of the system (25) shows that

\partial_{α} d = C_{α 3}^{τ} a_{τ} + C_{α 3}^{3} d in ω .

Hence,

\begin{matrix} a_{α} \cdot \partial_{β} d + a_{β} \cdot \partial_{α} d & = C_{β 3}^{τ} (a_{α} \cdot a_{τ}) + C_{β 3}^{3} (a_{α} \cdot d) \\ + C_{α 3}^{τ} (a_{β} \cdot a_{τ}) + C_{α 3}^{3} (a_{β} \cdot d) . \end{matrix}

Furthermore, using relations (31) in the right-hand side of the above equation implies that

a_{α} \cdot \partial_{β} d + a_{β} \cdot \partial_{α} d = C_{β 3}^{τ} a_{ατ} + C_{β 3}^{3} d_{α} + C_{α 3}^{τ} a_{βτ} + C_{α 3}^{3} d_{β} .

Replacing the functions $C_{β 3}^{τ}$ and $C_{α 3}^{τ}$ appearing in the right-hand side above by their expressions given by (32), we next deduce that

\begin{matrix} a_{α} \cdot \partial_{β} d + a_{β} \cdot \partial_{α} d & = (k_{βα} - \frac{d_{β | α} - d_{α | β}}{2}) - C_{β 3}^{3} d^{τ} a_{ατ} + C_{β 3}^{3} d_{α} \\ + (k_{αβ} - \frac{d_{α | β} - d_{β | α}}{2}) - C_{α 3}^{3} d^{τ} a_{βτ} + C_{α 3}^{3} d_{β} \\ = 2 k_{αβ}, \end{matrix}

as desired. Note that the last equality above holds since the matrix field $(k_{αβ})$ is symmetric by the assumptions of the theorem and since $d^{τ} a_{ατ} = d_{α}$ by the definition (18) of the functions $d^{β}$ appearing in the definition (17) of the functions $C_{αj}^{i}$ .

Finally, the pair $(φ, d)$ defines a surface with assigned director field according to Definition 1, as we now show.

The first equation of system (31) together with the assumption that the matrix $(a_{αβ})$ is positive-definite implies that the two vector fields

a_{α} : = \partial_{α} φ : ω \to ℝ^{3}

are linearly independent. Then, the unit vector field

a_{3} : = \frac{a_{1} \land a_{2}}{| a_{1} \land a_{2} |} : ω \to ℝ^{3}

is well defined and perpendicular to both vector fields $a_{α}$ , so the three vector fields ${a_{1}, a_{2}, a_{3}}$ form a basis of $ℝ^{3}$ at every point of ω. Let $(a^{αβ})$ denote the inverse of the matrix field $(a_{αβ})$ , and let

a^{α} : = a^{αβ} a_{β} and a^{3} : = a_{3} .

Then, one can prove that the three vectors ${a^{1}, a^{2}, a^{3}}$ form the dual basis of ${a_{1}, a_{2}, a_{3}}$ , that they satisfy

a^{α} \cdot a^{β} = a^{αβ} and a^{α} \cdot a^{3} = 0 in ω,

and that the vector field d admits the following decomposition over the basis ${a^{1}, a^{2}, a^{3}}$ of $ℝ^{3}$ :

d = (d \cdot a_{i}) a^{i} .

The second and third equations of system (31) next imply that

d = d_{α} a^{α} + (d \cdot a_{3}) a^{3}

and

n = | d |^{2} = a^{αβ} d_{β} d_{α} + {(d \cdot a_{3})}^{2} .

But

n - a^{αβ} d_{α} d_{β} > 0 in ω

by the assumptions of the theorem. Hence, the previous relation implies that

{(d \cdot a_{3})}^{2} > 0 in ω .

The vector fields d and $a_{3}$ being continuous over ω (since they belong to the space $W_{loc}^{1, p}$ with $p > 2$ in a two-dimensional domain), which is a connected open set, it follows that either $(d \cdot a_{3}) (y) > 0$ for all $y \in ω$ , or $(d \cdot a_{3}) (y) > 0$ for all $y \in ω$ . But we already proved that $(d \cdot a_{3}) (y_{0}) > 0$ for some $y_{0} \in ω$ (see the last relation of system (28)). Hence

(d \cdot a_{3}) > 0 in ω,

or equivalently, in view of the definition of the vector field $a_{3}$ ,

(\partial_{α} φ \land \partial_{β} φ) \cdot d > 0 in ω .

This inequality shows that $(φ, d)$ define a surface with assigned director field according to Definition 1.

This completes the proof of the existence part of the theorem.

To prove the uniqueness part of the theorem, let there be given another surface with assigned director field $(\tilde{φ}, \tilde{d})$ , with $\tilde{φ} : ω \to ℝ^{3}$ of class $W_{loc}^{2, p}$ and $\tilde{d} : ω \to ℝ^{3}$ of class $W_{loc}^{1, p}$ , satisfying the equations

\begin{matrix} a_{αβ} & = \partial_{α} \tilde{φ} \cdot \partial_{β} \tilde{φ}, \\ k_{αβ} & = \frac{1}{2} (\partial_{α} \tilde{φ} \cdot \partial_{β} \tilde{d} + \partial_{α} \tilde{d} \cdot \partial_{β} \tilde{φ}), \\ d_{α} & = \partial_{α} \tilde{φ} \cdot \tilde{d}, \\ n & = | \tilde{d} |^{2} . \end{matrix}

(33)

By Theorem 2, we know that the vector fields ${\tilde{a}}_{α} : = \partial_{α} \tilde{φ}$ and $\tilde{d}$ satisfy the system

\begin{matrix} \partial_{α} {\tilde{a}}_{β} & = {\tilde{C}}_{αβ}^{σ} {\tilde{a}}_{σ} + {\tilde{C}}_{αβ}^{3} \tilde{d} & in ω, \\ \partial_{α} \tilde{d} & = {\tilde{C}}_{α 3}^{σ} {\tilde{a}}_{σ} + {\tilde{C}}_{α 3}^{3} \tilde{d} & in ω, \end{matrix}

(34)

while the functions $a_{α} : = \partial_{α} φ$ and d satisfy the system

\begin{matrix} \partial_{α} a_{β} & = C_{αβ}^{σ} a_{σ} + C_{αβ}^{3} d & in ω, \\ \partial_{α} d & = C_{α 3}^{σ} a_{σ} + C_{α 3}^{3} d & in ω, \end{matrix}

(35)

where the functions ${\tilde{C}}_{αj}^{i}$ and $C_{αj}^{i}$ are defined by the same expressions (17) in terms of the functions $(a_{αβ}, k_{αβ}, d_{α}, n)$ , thanks to the systems (23) and (33) established above. Hence

{\tilde{C}}_{αj}^{i} = C_{αj}^{i} in ω,

so that the Pfaff systems satisfied by the vector fields $(a_{α}, d)$ and $({\tilde{a}}_{α}, \tilde{d})$ coincide.

Systems (23) and (33) also imply that the two matrix fields

F : = (\begin{matrix} a_{1} | a_{2} | d \end{matrix}) and \tilde{F} : = (\begin{matrix} {\tilde{a}}_{1} | {\tilde{a}}_{2} | \tilde{d} \end{matrix}),

whose column vector fields are the vector fields $(a_{α}, d)$ and $({\tilde{a}}_{α}, \tilde{d})$ , respectively, satisfy

F^{T} F = (\begin{matrix} a_{11} & a_{12} & d_{1} \\ a_{21} & a_{22} & d_{2} \\ d_{1} & d_{2} & n \end{matrix}) = {\tilde{F}}^{T} \tilde{F} in ω .

(36)

Besides, the matrix fields F and $\tilde{F}$ are invertible since we already proved that their column vector fields satisfy

(a_{1} \land a_{2}) \cdot d = (\partial_{1} φ \land \partial_{2} φ) \cdot d > 0 in ω,

and

({\tilde{a}}_{1} \land {\tilde{a}}_{2}) \cdot \tilde{d} = (\partial_{1} \tilde{φ} \land \partial_{2} \tilde{φ}) \cdot \tilde{d} > 0 in ω,

which is equivalent to

\det F > 0 in ω

and

\det \tilde{F} > 0 in ω,

respectively. This implies that the matrix field F is invertible and

\det (\tilde{F} F^{- 1}) > 0 in ω .

Besides, equation (36) implies that

{(\tilde{F} F^{- 1})}^{T} (\tilde{F} F^{- 1}) = I : = (\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) in ω .

We just proved that $(\tilde{F} F^{- 1})$ is a special orthogonal matrix at every point of ω.

Let $y_{0}$ be any point in ω and let $R : = (\tilde{F} F^{- 1}) (y_{0})$ . Then, R is a special orthogonal matrix such that

\begin{matrix} (R a_{α}) (y_{0}) & = {\tilde{a}}_{α} (y_{0}), \\ (R d) (y_{0}) & = \tilde{d} (y_{0}) . \end{matrix}

Besides, using that ${\tilde{C}}_{αj}^{i} = C_{αj}^{i}$ in system (34), on the one hand, and multiplying system (35) to the left by the matrix R , on the other hand, show that

\begin{matrix} \partial_{α} {\tilde{a}}_{β} & = C_{αβ}^{σ} {\tilde{a}}_{σ} + C_{αβ}^{3} \tilde{d} & in ω, \\ \partial_{α} \tilde{d} & = C_{α 3}^{σ} {\tilde{a}}_{σ} + C_{α 3}^{3} \tilde{d} & in ω, \end{matrix}

and

\begin{matrix} \partial_{α} (R a_{β}) & = C_{αβ}^{σ} (R a_{σ}) + C_{αβ}^{3} (R d) & in ω, \\ \partial_{α} (R d) & = C_{α 3}^{σ} (R a_{σ}) + C_{α 3}^{3} (R d) & in ω . \end{matrix}

Thus,

{\tilde{a}}_{β} = R a_{β} and \tilde{d} = R d in ω,

by the uniqueness theorem about Pfaff systems proven in S. Mardare [14, Theorem 4.2].

Furthermore, since ${\tilde{a}}_{β} = \partial_{α} \tilde{φ}$ and $a_{β} = \partial_{α} φ$ , the relation ${\tilde{a}}_{β} = R a_{β}$ established above implies that

\partial_{α} (\tilde{φ} - R φ) = 0 in ω .

Then, $(\tilde{φ} - R φ)$ is a constant mapping, since the set ω is connected by the assumptions of the theorem. Defining the vector $a : = (\tilde{φ} - R φ) (y_{0}) \in ℝ^{3}$ , we thus have

(\tilde{φ} - R φ) (y) = a for all y \in ω,

or equivalently

\tilde{φ} (y) = a + R φ (y) for all y \in ω .

This completes the proof of the uniqueness part of the theorem. □

5. The Gauss and Codazzi–Mainardi equations

We prove in this section that the compatibility conditions (22) established in Theorem 3 generalize the classical Gauss and Codazzi–Mainardi equations for surfaces (see, e.g., [15 –17]) to surfaces with assigned director field.

Theorem 5. Let ω be an open subset of $ℝ^{2}$ and let $p \in (2, + \infty]$ . Given any symmetric and positive-definite matrix field $(a_{αβ}) : ω \to ℝ^{2 \times 2}$ of class $W_{loc}^{1, p}$ , let $(a^{αβ})$ denotes the inverse of the matrix field $(a_{αβ})$ , and let

Γ_{αβ, σ} : = \frac{1}{2} (\partial_{α} a_{βσ} + \partial_{β} a_{ασ} - \partial_{σ} a_{αβ}) and Γ_{αβ}^{τ} : = a^{στ} Γ_{αβ, σ} .

Let a vector field $(d_{α}) : ω \to ℝ^{2}$ of class $W_{loc}^{1, p}$ and a scalar field $n : ω \to ℝ$ of class $W_{loc}^{1, p}$ be given that together satisfy the condition

(n - a^{αβ} d_{α} d_{β}) (y) > 0 for all y \in ω .

Given in addition any symmetric matrix $(k_{αβ}) : ω \to ℝ^{2 \times 2}$ of class $L_{loc}^{p}$ , let the functions $C_{αj}^{i}$ and $ℓ_{αβ}$ be defined by relations (17) and Corollary 1, respectively, in terms of the functions $(a_{αβ}, k_{αβ}, d_{α}, n)$ .

Then, the matrix fields $C_{α} : = (C_{αj}^{i})$ satisfy the compatibility conditions of Theorem 3, viz.,

\partial_{α} C_{β} + C_{α} C_{β} = \partial_{β} C_{α} + C_{β} C_{α} in ω,

if and only if the functions $(a_{αβ}, ℓ_{αβ})$ defined in Corollary 1 in terms of the functions $(a_{αβ}, k_{αβ}, d_{α}, n)$ satisfy the following Gauss and Codazzi–Mainardi compatibility conditions:

\begin{matrix} \partial_{α} Γ_{βγ}^{τ} - \partial_{β} Γ_{αγ}^{τ} + Γ_{βγ}^{σ} Γ_{ασ}^{τ} - Γ_{αγ}^{σ} Γ_{βσ}^{τ} & = a^{στ} ℓ_{ασ} ℓ_{βγ} - a^{στ} ℓ_{βσ} ℓ_{αγ} & in ω, \\ \partial_{α} ℓ_{βγ} + Γ_{βγ}^{σ} ℓ_{ασ} & = \partial_{β} ℓ_{αγ} + Γ_{αγ}^{σ} ℓ_{βσ} & in ω, \end{matrix}

or equivalently

\begin{matrix} \partial_{α} Γ_{βγ, τ} - \partial_{β} Γ_{αγ, τ} + Γ_{αγ}^{σ} Γ_{βτ, σ} - Γ_{βγ}^{σ} Γ_{ατ, σ} & = ℓ_{ατ} ℓ_{βγ} - ℓ_{βτ} ℓ_{αγ} & in ω, \\ \partial_{α} ℓ_{βγ} + Γ_{βγ}^{σ} ℓ_{ασ} & = \partial_{β} ℓ_{αγ} + Γ_{αγ}^{σ} ℓ_{βσ} & in ω . \end{matrix}

Proof. Note that the functions $C_{αj}^{i}$ , $Γ_{αβ, σ}$ , $Γ_{αβ}^{τ}$ , and $ℓ_{αβ}$ belong to the space $L_{loc}^{p} (ω)$ , so their derivatives are well defined in the sense of distributions.

(i). Assume first that the matrix fields $C_{α}$ satisfy the compatibility conditions

\partial_{α} C_{β} + C_{α} C_{β} = \partial_{β} C_{α} + C_{β} C_{α} in ω .

Let $ω_{c}$ be any simply connected open subset of ω. First, Theorem 4 shows that there exists a surface with assigned director field $(φ, d)$ with $φ : ω_{c} \to ℝ^{3}$ of class $W_{loc}^{2, p}$ and $d : ω_{c} \to ℝ^{3}$ of class $W_{loc}^{1, p}$ such that

\begin{matrix} a_{αβ} & = \partial_{α} φ \cdot \partial_{β} φ & in ω_{c}, \\ k_{αβ} & = \frac{1}{2} (\partial_{α} φ \cdot \partial_{β} d + \partial_{α} d \cdot \partial_{β} φ) & in ω_{c}, \\ d_{α} & = \partial_{α} φ \cdot d and n = | d |^{2} & in ω_{c} . \end{matrix}

In particular, the first relation in the above system shows that the functions $a_{αβ}$ are the covariant components of the first fundamental form of the surface $S = φ (ω_{c})$ .

Second, Theorem 1 applied to the surface with assigned director field $(φ, d) : ω_{c} \to ℝ^{3} \times ℝ^{3}$ shows that (see (13))

ℓ_{αβ} = \partial_{αβ} φ \cdot a_{3} in ω_{c}, where a_{3} : = \frac{\partial_{1} φ \land \partial_{2} φ}{| \partial_{1} φ \land \partial_{2} φ |} .

This means that the functions $ℓ_{αβ}$ are the covariant components of the second fundamental form of the surface $S = φ (ω_{c})$ . That the functions $a_{αβ}$ and $ℓ_{αβ}$ must then satisfy the Gauss and Codazzi–Mainardi equations can then be proved by adapting the proof for smooth surfaces (see, e.g., Ciarlet [19]) to the case studied here of surfaces defined by immersions $φ : ω_{c} \to ℝ^{3}$ of class $W_{loc}^{2, p}$ , $p > 2$ , only.

More specifically, since the functions $a_{αβ}$ and $ℓ_{αβ}$ are the covariant components of respectively the first and second fundamental forms of the surface $S = φ (ω_{c})$ , the vector fields $a_{α} : = \partial_{α} φ$ and $a_{3}$ satisfy the following Gauss and Weingarten equations (see system (6))

\begin{matrix} \partial_{α} a_{β} & = Γ_{αβ}^{σ} a_{σ} + ℓ_{αβ} a_{3} & a.e. in ω_{c}, \\ \partial_{α} a_{3} & = - a^{σβ} ℓ_{αβ} a_{σ} & a.e. in ω_{c} . \end{matrix}

Since the vector fields $a_{γ}$ and $a_{3}$ belong to the space $W_{loc}^{1, p} (ω_{c}; ℝ^{3})$ with $p > 2$ , the second derivatives of $a_{γ}$ (in particular) are well defined in the space of distributions, and they satisfy (note that the products of functions by distributions appearing in the next calculations are well defined in the space of distributions thanks to the assumption that $p > 2$ , which implies that the space $W_{loc}^{1, p} (ω)$ is an algebra embedded in the space $C_{b}^{0} (ω)$ by the Sobolev embedding theorem):

\begin{matrix} \partial_{α} (\partial_{β} a_{γ}) & = \partial_{α} (Γ_{βγ}^{τ} a_{τ} + ℓ_{βγ} a_{3}) \\ = \partial_{α} Γ_{βγ}^{τ} a_{τ} + Γ_{βγ}^{τ} \partial_{α} a_{τ} + \partial_{α} ℓ_{βγ} a_{3} + ℓ_{βγ} \partial_{α} a_{3} \\ = \partial_{α} Γ_{βγ}^{σ} a_{σ} + Γ_{βγ}^{τ} (Γ_{ατ}^{σ} a_{σ} + ℓ_{ατ} a_{3}) + \partial_{α} ℓ_{βγ} a_{3} - ℓ_{βγ} (a^{στ} ℓ_{ατ} a_{σ}) \\ = (\partial_{α} Γ_{βγ}^{σ} + Γ_{βγ}^{τ} Γ_{ατ}^{σ} - ℓ_{βγ} a^{στ} ℓ_{ατ}) a_{σ} + (\partial_{α} ℓ_{βγ} + Γ_{βγ}^{τ} ℓ_{ατ}) a_{3} . \end{matrix}

Since the vector fields ${a_{1}, a_{2}, a_{3}}$ are linearly independent at every point of ω (this is a consequence of the assumption that the symmetric matrix field $(a_{αβ})$ is positive-definite at every point of ω), the symmetry of the second derivatives of the vector fields $a_{γ}$ , viz.,

\partial_{α} (\partial_{β} a_{γ}) = \partial_{β} (\partial_{α} a_{γ}) in D^{'} (ω_{c}; ℝ^{3}),

implies that the following relations hold in $D^{'} (ω_{c})$ :

\begin{matrix} \partial_{α} Γ_{βγ}^{σ} + Γ_{βγ}^{τ} Γ_{ατ}^{σ} - ℓ_{βγ} a^{στ} ℓ_{ατ} & = \partial_{β} Γ_{αγ}^{σ} + Γ_{αγ}^{τ} Γ_{βτ}^{σ} - ℓ_{αγ} a^{στ} ℓ_{βτ}, \\ \partial_{α} ℓ_{βγ} + Γ_{βγ}^{τ} ℓ_{ατ} & = \partial_{β} ℓ_{αγ} + Γ_{αγ}^{τ} ℓ_{βτ} . \end{matrix}

Since these relations hold for all simply connected open subsets $ω_{c}$ of ω, they also hold in $D^{'} (ω)$ . Thus, the Gauss and Codazzi–Mainardi equations stated in the theorem are satisfied.

(ii). Assume now that the functions $(a_{αβ}, ℓ_{αβ})$ defined in Corollary 1 in terms of the functions $(a_{αβ}, k_{αβ}, d_{α}, n)$ satisfy the Gauss–Codazzi compatibility conditions stated in the theorem. Let $ω_{c}$ be any simply connected open subset of ω.

Then, the fundamental theorem of surface theory in its general form due to S. Mardare [20] shows that there exists an immersion $φ : ω_{c} \to ℝ^{3}$ of class $W_{loc}^{2, p}$ such that

\begin{matrix} a_{αβ} & = \partial_{α} φ \cdot \partial_{β} φ & in ω_{c}, \\ ℓ_{αβ} & = \partial_{αβ} φ \cdot a_{3} & a.e. in ω_{c}, \end{matrix}

where

a_{3} : = \frac{\partial_{1} φ \land \partial_{2} φ}{| \partial_{1} φ \land \partial_{2} φ |} .

Then, the vector fields $a_{α} : = \partial_{α} φ$ and $a_{3}$ satisfy the Gauss and Weingarten equations (see system (6))

\begin{matrix} \partial_{α} a_{β} & = Γ_{αβ}^{σ} a_{σ} + ℓ_{αβ} a_{3} & a.e. in ω_{c}, \\ \partial_{α} a_{3} & = - a^{σβ} ℓ_{αβ} a_{σ} & a.e. in ω_{c} . \end{matrix}

Since $(n - a^{αβ} d_{α} d_{β}) > 0$ at all points of ω by the assumptions of the theorem, the function $d_{3} : = {(n - a^{αβ} d_{α} d_{β})}^{1 ∕ 2}$ is well defined. Let $d^{α} : = a^{αβ} d_{β}$ . Then, the vector field

d : = d^{σ} a_{σ} + d_{3} a_{3}

is well defined, it belongs to the space $W_{loc}^{1, p} (ω; ℝ^{3})$ , and we have

a_{3} = \frac{1}{d_{3}} d - \frac{d^{σ}}{d_{3}} a_{σ} .

Using this expression of $a_{3}$ in the Gauss and Weingarten equations above implies that

\begin{matrix} \partial_{α} a_{β} = (Γ_{αβ}^{σ} - \frac{d^{σ}}{d_{3}} ℓ_{αβ}) a_{σ} + (\frac{1}{d_{3}} ℓ_{αβ}) d, \\ \frac{1}{d_{3}} (\partial_{α} d - \partial_{α} d^{σ} a_{σ} - d^{β} \partial_{α} a_{β}) - \frac{\partial_{α} d_{3}}{{(d_{3})}^{2}} (d - d^{σ} a_{σ}) = - a^{σβ} ℓ_{αβ} a_{σ} . \end{matrix}

Substituting $\partial_{α} a_{β}$ in the second equation by its expression given in the first one yields

\begin{matrix} \partial_{α} d & = (- d_{3} a^{σβ} ℓ_{αβ} + \partial_{α} d^{σ} + d^{β} (Γ_{αβ}^{σ} - \frac{d^{σ}}{d_{3}} ℓ_{αβ}) - \frac{d^{σ} \partial_{α} d_{3}}{d_{3}}) a_{σ} \\ + \frac{1}{d_{3}} (\partial_{α} d_{3} + d^{β} ℓ_{αβ}) d, \end{matrix}

so that the previous system becomes

\begin{matrix} \partial_{α} a_{β} & = (Γ_{αβ}^{σ} - \frac{d^{σ}}{d_{3}} ℓ_{αβ}) a_{σ} + (\frac{1}{d_{3}} ℓ_{αβ}) d, \\ \partial_{α} d & = (- d_{3} ℓ_{α}^{σ} + d^{σ} |_{α} - \frac{d^{σ}}{d_{3}} (\partial_{α} d_{3} + d^{β} ℓ_{αβ})) a_{σ} + \frac{1}{d_{3}} (\partial_{α} d_{3} + d^{β} ℓ_{αβ}) d, \end{matrix}

where $ℓ_{α}^{τ} : = a^{τσ} ℓ_{ασ}$ and $d^{τ} |_{α} : = \partial_{α} d^{τ} + Γ_{ασ}^{τ} d^{σ}$ .

Observe that the coefficients of the above system are precisely the functions $C_{αj}^{i}$ defined by relations (17), thanks to Corollary 1. Hence, the above system is equivalent to the system

\begin{matrix} \partial_{α} a_{β} & = C_{αβ}^{σ} a_{σ} + C_{αβ}^{3} d in ω_{c}, \\ \partial_{α} d & = C_{α 3}^{σ} a_{σ} + C_{α 3}^{3} d in ω_{c}, \end{matrix}

which itself is equivalent to the Pfaff system

\partial_{α} F = F C_{α} in ω_{c},

where F denotes the matrix field with $\partial_{1} φ$ , $\partial_{2} φ$ , and d as its column vectors (in this order) and $C_{α} : = (C_{αj}^{i})$ . Then, its coefficients necessarily satisfy the compatibility condition

\partial_{α} C_{β} + C_{α} C_{β} = \partial_{β} C_{α} + C_{β} C_{α} in ω_{c},

as proven by S. Mardare [14]. Since this relation holds for all simply connected open subsets $ω_{c}$ of ω, it also holds in ω. □

Remark 5. The symmetry of the second derivatives defined in the sense of distributions implies that there are only one independent Gauss equation, obtained for $α = γ = 1$ and $β = τ = 2$ in the first equation of the Gauss–Codazzi system in Theorem 5, and only two Codazzi (also called Codazzi–Mainardi) equations, one obtained for $α = γ = 1$ and $β = 2$ , and the other obtained for $α = 1$ and $β = γ = 2$ in the second equation of the same system.

6. Continuity of the surface with assigned director field as a function of its invariants $(a_{α β}, k_{α β}, d_{α, n})$

We showed in the previous section that a surface with assigned director field $(φ, d)$ , with $φ : ω \to ℝ^{3}$ of class $W_{loc}^{2, p}$ and $d : ω \to ℝ^{3}$ of class $W_{loc}^{1, p}$ , can be recovered from the tensor fields

a_{αβ} \in W_{loc}^{1, p} (ω), k_{αβ} \in L_{loc}^{p} (ω), d_{α} \in W_{loc}^{1, p} (ω), and n \in W_{loc}^{1, p} (ω)

(provided these tensors satisfy the assumptions of Theorem 4), and that the reconstructed surface with assigned director field is unique up to an isometry of the three-dimensional space in which this surface is immersed.

We prove in this section that the mapping

(a_{αβ}, k_{αβ}, d_{α}, n) \to (φ, d)

defined in this fashion is continuous when the above spaces are equipped with their natural topologies (see Theorem 7 below). This theorem generalizes previous results about the continuity of the mapping $(a_{αβ}, b_{αβ}) \to φ$ associated with classical surfaces in $ℝ^{3}$ ; in this respect, see [1, 5 –7, 11, 13, 21 –23].

The proof of the continuity result mentioned above relies on Theorem 2 established in this paper and on a stability result for Pfaff systems with coefficients in $L^{p} (ω)$ established in Ciarlet and S. Mardare [24]. We state this stability result in the next theorem for reader’s convenience.

We remind that an open subset ω of $ℝ^{2}$ satisfies the uniform interior cone property if there exists a bounded open cone Δ in $ℝ^{2}$ with vertex at the origin $0 \in ℝ^{2}$ such that, for each point $y \in ω$ , there exists a rotation $R : ℝ^{2} \to ℝ^{2}$ such that $(y + R Δ) ⋐ ω$ (see, e.g., Adams and Fournier [25] for more details about this notion). Note that, in particular, a bounded Lipschitz domain satisfies the uniform interior cone property.

In all that follows, a notation such as $C = C (ω, y_{0}, p, m, M)$ means that C is a constant that depends on the parameters $ω, y_{0}, p$ , m, and M (and only on them).

Theorem 6. Let ω be a bounded and connected open subset of $ℝ^{2}$ that satisfies the uniform interior cone property, let $y_{0} \in ω$ , let $p > 2$ , let $n \in ℕ^{*}$ , and let $M > 0$ . Then, there exists a constant $C = C (ω, y_{0}, p, n, M)$ such that

∥ \tilde{F} - F ∥_{W^{1, p} (ω)} \leq C {| (\tilde{F} - F) (y_{0}) | + \sum_{α} ∥ {\tilde{C}}_{α} - C_{α} ∥_{L^{p} (ω)}}

for all matrix fields $F, \tilde{F} \in W^{1, p} (ω; ℝ^{n \times n})$ and $C_{α}, {\tilde{C}}_{α} \in L^{p} (ω; ℝ^{n \times n})$ that satisfy the Pfaff systems

\partial_{α} F = F C_{α} and \partial_{α} \tilde{F} = \tilde{F} {\tilde{C}}_{α} in ω,

together with the estimates

| F (y_{0}) | \leq M and | \tilde{F} (y_{0}) | \leq M,

and

\sum_{α} {∥ C_{α} ∥}_{L^{p} (ω)} \leq M and \sum_{α} ∥ {\tilde{C}}_{α} ∥_{L^{p} (ω)} \leq M .

Proof. The proof is similar to that of Theorem 3.1 in Ciarlet & S. Mardare [24]. □

We are now in a position to prove the continuity result mentioned in the introduction to this section. It is an immediate consequence of the estimate (39) of the following theorem:

Theorem 7. Given any bounded and connected open set $ω \subset ℝ^{2}$ that satisfies the uniform interior cone property, any point $y_{0} \in ω$ , and any numbers $p \in (2, + \infty]$ and $δ \in (0, 1]$ , there exists a constant $C = C (ω, y_{0}, p, δ)$ with the following property:

For each pair of surfaces with assigned director field $(φ, d)$ and $(\tilde{φ}, \tilde{d})$ such that $φ, \tilde{φ} \in W^{2, p} (ω; ℝ^{3})$ , $d, \tilde{d} \in W^{1, p} (ω; ℝ^{3})$ ,

\begin{matrix} (\partial_{1} φ \land \partial_{2} φ) \cdot d \geq δ in ω, \\ ∥ \partial_{α} φ ∥_{W^{1, p} (ω)} \leq δ^{- 1} and ∥ d ∥_{W^{1, p} (ω)} \leq δ^{- 1}, \end{matrix}

(37)

and

\begin{matrix} (\partial_{1} \tilde{φ} \land \partial_{2} \tilde{φ}) \cdot \tilde{d} \geq δ in ω, \\ ∥ \partial_{α} \tilde{φ} ∥_{W^{1, p} (ω)} \leq δ^{- 1} and ∥ \tilde{d} ∥_{W^{1, p} (ω)} \leq δ^{- 1}, \end{matrix}

(38)

there exists a special orthogonal matrix $R_{0} = R_{0} (\partial_{α} φ (y_{0}), \partial_{α} \tilde{φ} (y_{0}))$ and a vector $a_{0} = a_{0} (φ (y_{0}),$ $\tilde{φ} (y_{0}), \partial_{α} φ (y_{0}), \partial_{α} \tilde{φ} (y_{0}))$ such that

\begin{matrix} ∥ \tilde{φ} - (a_{0} + R_{0} φ) ∥_{W^{2, p} (ω)} + ∥ \tilde{d} - R_{0} d ∥_{W^{1, p} (ω)} \\ \leq C {\sum_{α, β} ∥ {\tilde{a}}_{αβ} - a_{αβ} ∥_{W^{1, p} (ω)} + \sum_{α, β} ∥ {\tilde{k}}_{αβ} - k_{αβ} ∥_{L^{p} (ω)} \\ + \sum_{α} ∥ {\tilde{d}}_{α} - d_{α} ∥_{W^{1, p} (ω)} + ∥ \tilde{n} - n ∥_{W^{1, p} (ω)}}, \end{matrix}

(39)

where the functions appearing in the right-hand side of (39) are defined by system (16) in terms of either $(\tilde{φ}, \tilde{d})$ or $(φ, d)$ , depending on whether they are denoted by letters surmounted, or not surmounted, by a tilde.

Proof. The proof consists of the following five steps, numbered (i) to (v).

(i) Definition of the special orthogonal matrix $R_{0}$ and of the vector $a_{0}$ appearing in inequality (39) of the theorem.

Let $F : = (\begin{matrix} \partial_{1} φ | \partial_{2} φ | d \end{matrix})$ be the matrix field with columns $\partial_{1} φ$ , $\partial_{2} φ$ , and d , and let $G : ω \to ℝ^{3 \times 3}$ be the matrix field defined by

G : = (\begin{matrix} r_{11} & r_{12} & r_{1 α} a^{αβ} d_{β} \\ r_{21} & r_{22} & r_{2 α} a^{αβ} d_{β} \\ 0 & 0 & {(n - a^{στ} d_{σ} d_{τ})}^{1 ∕ 2} \end{matrix}),

(40)

where the functions $a_{αβ}$ , $d_{α}$ , and n, are defined by relations (16) in terms of $(φ, d)$ , and where $(r_{αβ}) : = {(a_{αβ})}^{1 ∕ 2}$ and $(a^{αβ}) : = {(a_{αβ})}^{- 1}$ .

Note that the matrix field G is well defined thanks to the assumption (37) of the theorem. Indeed, the first inequality of (37) implies that the two vector fields ${\partial_{1} φ, \partial_{2} φ}$ are linearly independent at every point of ω and that (the first equality below has been established in the proof of Corollary 3)

\begin{matrix} n - a^{στ} d_{σ} d_{τ} & = \frac{{((\partial_{1} φ \land \partial_{2} φ) \cdot d)}^{2}}{| \partial_{1} φ \land \partial_{2} φ |^{2}} > 0 in ω . \end{matrix}

Thus, the matrix field $(a_{αβ}) : ω \to ℝ^{2 \times 2}$ is symmetric and positive-definite, which in turn implies that its square root $(r_{αβ})$ and its inverse $(a^{αβ})$ are well-defined symmetric and positive-definite matrix fields.

Note also that the matrix fields F and G have positive determinants, so they are invertible at every point of ω, since

\det F = (\partial_{1} φ \land \partial_{2} φ) \cdot d \geq δ > 0 in ω,

and

\det G = {(n - a^{στ} d_{σ} d_{τ})}^{1 ∕ 2} \det (r_{αβ}) > 0 in ω,

thanks to assumption (37).

Then, let the matrix fields $\tilde{F}$ and $\tilde{G}$ be defined in a similar fashion, but in terms of $(\tilde{φ}, \tilde{d})$ instead of $(φ, d)$ . Obviously, they satisfy similar properties as those established above for F and G ; in particular, they have positive determinants and are invertible at every point of ω.

Finally, define the matrix and vector fields $R : ω \to ℝ^{3 \times 3}$ and $a : ω \to ℝ^{3}$ by

\begin{matrix} R & : = \tilde{F} {\tilde{G}}^{- 1} G^{- T} F^{T}, \\ a & : = \tilde{φ} - R φ . \end{matrix}

(41)

We will prove that inequality (39) holds with

a_{0} : = a (y_{0}) and R_{0} = R (y_{0}) .

(42)

Proving that $R_{0}$ is a special orthogonal matrix.

First, $\det R > 0$ at every point of ω since R is a product of matrix fields with positive determinants. Second, $R^{T} R = I$ in ω, where $I : ω \to ℝ^{3 \times 3}$ denotes the identity matrix field, as we now show. Noting that the definition of the matrix fields F and G implies that

F^{T} F = A : = (\begin{matrix} a_{11} & a_{12} & d_{1} \\ a_{21} & a_{22} & d_{2} \\ d_{1} & d_{2} & n \end{matrix}) in ω,

(43)

and

G^{T} G = A in ω,

(44)

we deduce that

F {(G^{T} G)}^{- 1} F^{T} = F {(F^{T} F)}^{- 1} F^{T} = I in ω,

on the one hand. On the other hand, a similar argument applied to the matrix fields $\tilde{F}$ and $\tilde{G}$ shows that they satisfy

{\tilde{F}}^{T} \tilde{F} = {\tilde{G}}^{T} \tilde{G} = \tilde{A} : = (\begin{matrix} {\tilde{a}}_{11} & {\tilde{a}}_{12} & {\tilde{d}}_{1} \\ {\tilde{a}}_{21} & {\tilde{a}}_{22} & {\tilde{d}}_{2} \\ {\tilde{d}}_{1} & {\tilde{d}}_{2} & \tilde{n} \end{matrix}) in ω,

which in turn implies that

{\tilde{G}}^{- T} ({\tilde{F}}^{T} \tilde{F}) {\tilde{G}}^{- 1} = {\tilde{G}}^{- T} ({\tilde{G}}^{T} \tilde{G}) {\tilde{G}}^{- 1} = I in ω .

Hence,

\begin{matrix} R^{T} R & = F G^{- 1} [{\tilde{G}}^{- T} ({\tilde{F}}^{T} \tilde{F}) {\tilde{G}}^{- 1}] G^{- T} F^{T} = F G^{- 1} G^{- T} F^{T} \\ = I in ω . \end{matrix}

(ii) Estimates of the functions describing the geometry of the surface with assigned director field $(φ, d)$ .

Since $p > 2$ and ω is a bounded and connected open set in $ℝ^{2}$ satisfying the uniform interior cone property, the space $W^{1, p} (ω)$ is an algebra that can be identified with a subspace of the space of bounded continuous functions $C_{b}^{0} (ω)$ by the Sobolev embedding theorem (see, e.g., Adams & Fournier [25]). Hence, there exists a constant $C = C (ω, p) > 0$ such that, for all functions $f, g \in W^{1, p} (ω)$ ,

\begin{matrix} ∥ fg ∥_{W^{1, p} (ω)} & \leq C ∥ f ∥_{W^{1, p} (ω)} ∥ g ∥_{W^{1, p} (ω)}, \\ ∥ f ∥_{L^{\infty} (ω)} & \leq C ∥ f ∥_{W^{1, p} (ω)} \end{matrix}

In what follows, we will use the same notation $C > 0$ for different positive constants.

The assumptions of the theorem imply, in particular, that the functions $a_{αβ}$ , $d_{α}$ , and n belong to $W^{1, p} (ω)$ , that the function $k_{αβ}$ belongs to $L^{p} (ω)$ , and that

\begin{matrix} ∥ a_{αβ} ∥_{W^{1, p} (ω)} \leq C δ^{- 2}, & ∥ k_{αβ} ∥_{L^{p} (ω)} \leq C δ^{- 2}, \\ ∥ d_{α} ∥_{W^{1, p} (ω)} \leq C δ^{- 2}, & ∥ n ∥_{W^{1, p} (ω)} \leq C δ^{- 2} . \end{matrix}

Moreover, the first inequality of system (37) implies that the vector fields $a_{α} : = \partial_{α} φ$ and d satisfy

| a_{1} \land a_{2} | | d | \geq | (a_{1} \land a_{2}) \cdot d | \geq δ in ω,

which combined with the second inequalities of the same system next implies that

| a_{1} \land a_{2} | \geq C δ^{2} in ω,

then that

\det (a_{αβ}) = \det (a_{α} \cdot a_{β}) = | a_{1} \land a_{2} |^{2} \geq C δ^{4} in ω .

Therefore, the inverse matrix field

(a^{αβ}) : = {(a_{αβ})}^{- 1} = \frac{1}{\det (a_{αβ})} (\begin{matrix} a_{22} & - a_{12} \\ - a_{21} & a_{11} \end{matrix})

(45)

is well defined, and its components satisfy $a^{αβ} \in W^{1, p} (ω)$ . Besides,

\begin{matrix} ∥ a^{αβ} ∥_{L^{\infty} (ω)} & \leq C δ^{- 6}, \\ ∥ a^{αβ} ∥_{W^{1, p} (ω)} & \leq C δ^{- 14} . \end{matrix}

Furthermore, the functions

Γ_{αβ}^{τ} : = \frac{1}{2} a^{στ} (\partial_{α} a_{βσ} + \partial_{β} a_{ασ} - \partial_{σ} a_{αβ})

belong to the space $L^{p} (ω)$ , and they satisfy

∥ Γ_{αβ}^{τ} ∥_{L^{p} (ω)} \leq C δ^{- 8} .

The function $(n - a^{στ} d_{σ} d_{τ})$ belongs to the space $W^{1, p} (ω)$ and satisfies (the first equality below has been established in the proof of Corollary 3)

\begin{matrix} n - a^{στ} d_{σ} d_{τ} & = \frac{{((\partial_{1} φ \land \partial_{2} φ) \cdot d)}^{2}}{| \partial_{1} φ \land \partial_{2} φ |^{2}} \geq C δ^{6} in ω . \end{matrix}

Consequently, the function ${(n - a^{στ} d_{σ} d_{τ})}^{- 1}$ is well defined, it belongs to the space $L^{\infty} (ω)$ , and its norm is bounded above by

{∥ {(n - a^{στ} d_{σ} d_{τ})}^{- 1} ∥}_{L^{\infty} (ω)} \leq C δ^{- 6} .

The functions $d_{α}$ and $d^{τ} : = a^{τα} d_{α}$ belong to the space $W^{1, p} (ω)$ , hence to the space $L^{\infty} (ω)$ by Sobolev embedding theorem, and they possess the following upper bounds:

\begin{matrix} ∥ d_{α} ∥_{W^{1, p} (ω)} = ∥ d \cdot \partial_{α} φ ∥_{W^{1, p} (ω)} \leq C δ^{- 2}, \\ ∥ d_{α} ∥_{L^{\infty} (ω)} = ∥ d \cdot \partial_{α} φ ∥_{L^{\infty} (ω)} \leq C δ^{- 2}, \\ ∥ d^{α} ∥_{L^{\infty} (ω)} = ∥ a^{αβ} d_{β} ∥_{L^{\infty} (ω)} \leq C δ^{- 8}, \\ ∥ d^{α} ∥_{W^{1, p} (ω)} = ∥ a^{αβ} d_{β} ∥_{W^{1, p} (ω)} \leq C δ^{- 16}, \\ ∥ d_{α} |_{β} ∥_{L^{p} (ω)} = ∥ \partial_{β} d_{α} - Γ_{αβ}^{τ} d_{τ} ∥_{L^{p} (ω)} \leq C δ^{- 10} . \end{matrix}

Using all the above estimates in the definition (17) of the functions $C_{αj}^{i}$ implies that there exists a constant $C = C (ω, p) \in ℝ$ such that

∥ C_{αj}^{i} ∥_{L^{p} (ω)} \leq C δ^{- 32} .

(46)

Finally, the matrix field F defined in part (i) of the proof and assumptions (37) of the theorem together imply that (remember that |⋅| denotes either the Frobenius norm of matrices or the Euclidean norm of vectors)

\begin{matrix} ∥ | F | ∥_{L^{\infty} (ω)} & \leq \sum_{α} {∥ | \partial_{α} φ | ∥}_{L^{\infty} (ω)} + {∥ | d | ∥}_{L^{\infty} (ω)} \\ \leq C (\sum_{α} {∥ \partial_{α} φ ∥}_{W^{1, p} (ω)} + {∥ d ∥}_{W^{1, p} (ω)}) \\ \leq C δ^{- 1} . \end{matrix}

(47)

Thus, using the invariance of Frobenius norm with respect to the multiplication by orthogonal matrices, we get

| (R F) (y_{0}) | = | F (y_{0}) | \leq C δ^{- 1} .

(iii). Estimates of the functions describing the geometry of the surface with assigned director field $(\tilde{φ}, \tilde{d})$ .

Similar estimates to those established in part (ii) of the proof hold for $(\tilde{φ}, \tilde{d})$ , since these fields satisfy the same assumptions as $(φ, d)$ . In particular,

∥ {\tilde{C}}_{αj}^{i} ∥_{L^{p} (ω)} \leq C_{1} δ^{- 32} .

and

| \tilde{F} (y_{0}) | \leq 2 C_{0} δ^{- 1} .

(48)

(iv). Estimate of the $W^{1, p} (ω)$ -norm of the matrix field $(\tilde{F} - R_{0} F)$ .

Let the functions ${\tilde{C}}_{αj}^{i}$ be defined by (17) in terms of the functions ${\tilde{a}}^{αβ}$ , ${\tilde{k}}_{αβ}$ , ${\tilde{d}}_{α}$ , and $\tilde{n}$ , the latter functions being themselves defined by (16) in terms of $(\tilde{φ}, \tilde{d})$ . Then, Theorem 2 shows that

\begin{matrix} \partial_{α} (\partial_{β} \tilde{φ}) & = C_{αβ}^{σ} \partial_{σ} \tilde{φ} + C_{αβ}^{3} \tilde{d} & in ω, \\ \partial_{α} \tilde{d} & = C_{α 3}^{σ} \partial_{σ} \tilde{φ} + C_{α 3}^{3} \tilde{d} & in ω, \end{matrix}

which means that the matrix field $\tilde{F}$ , defined in part (i) of the proof, satisfies the Pfaff system

\partial_{α} \tilde{F} = \tilde{F} {\tilde{C}}_{α} in ω,

(49)

where ${\tilde{C}}_{α} : = ({\tilde{C}}_{αj}^{i})$ is the matrix field with the function ${\tilde{C}}_{αj}^{i}$ at its i-th row and j-th column. Note that the assumptions of the theorem imply that $\tilde{F} \in W^{1, p} (ω; ℝ^{3 \times 3})$ and ${\tilde{C}}_{α} : = ({\tilde{C}}_{αj}^{i}) \in L^{p} (ω; ℝ^{3 \times 3})$ .

Likewise, we deduce that the matrix field F defined in part (i) of the proof satisfies the Pfaff system

\partial_{α} F = F C_{α} in ω,

where $C_{α} : = (C_{αj}^{i})$ is the matrix field formed by the functions $C_{αj}^{i}$ defined by (17) in terms of the functions $a^{αβ}$ , $k_{αβ}$ , $d_{α}$ , and n, which themselves are defined by (16) in terms of $(φ, d)$ . Multiplying this system on the left by the matrix $R_{0} : = R (y_{0})$ defined by (42) shows that the matrix field $(R_{0} F)$ satisfies the Pfaff system

\partial_{α} (R_{0} F) = (R_{0} F) C_{α} in ω .

(50)

Note that the assumptions of the theorem imply that $(R_{0} F) \in W^{1, p} (ω; ℝ^{3 \times 3})$ and $C_{α} : = (C_{αj}^{i}) \in L^{p} (ω; ℝ^{3 \times 3})$ .

We proved in parts (ii) and (iii) of the proof that the matrix fields appearing in the Pfaff systems (49) and (50) satisfy the estimates (46) to (48). This allows to apply Theorem 6 to these Pfaff systems and deduce in this way that

∥ \tilde{F} - R_{0} F ∥_{W^{1, p} (ω)} \leq C {| (\tilde{F} - R F) (y_{0}) | + \sum_{α} ∥ {\tilde{C}}_{α} - C_{α} ∥_{L^{p} (ω)}},

(51)

for some constant $C = C (ω, y_{0}, p, δ) \in ℝ$ . It remains to estimate the two norms appearing in the right-hand side of (51).

We begin by estimating the norm $| (\tilde{F} - R F) (y_{0}) |$ . First, we infer from relations (43) and (44) that

F^{T} F = G^{T} G = A : = (\begin{matrix} a_{11} & a_{12} & d_{1} \\ a_{21} & a_{22} & d_{2} \\ d_{1} & d_{2} & n \end{matrix}) in ω,

(52)

where the functions $a_{αβ}$ , $d_{α}$ , and n, are defined by (16) in terms of $(φ, d)$ , and, similarly,

{\tilde{F}}^{T} \tilde{F} = {\tilde{G}}^{T} \tilde{G} = \tilde{A} : = (\begin{matrix} {\tilde{a}}_{11} & {\tilde{a}}_{12} & {\tilde{d}}_{1} \\ {\tilde{a}}_{21} & {\tilde{a}}_{22} & {\tilde{d}}_{2} \\ {\tilde{d}}_{1} & {\tilde{d}}_{2} & \tilde{n} \end{matrix}) in ω .

(53)

Besides, the inverse of the matrix field A is given by

A^{- 1} = (\begin{matrix} a^{11} & a^{12} & 0 \\ a^{21} & a^{22} & 0 \\ 0 & 0 & 0 \end{matrix}) + \frac{1}{n - a^{στ} d_{σ} d_{τ}} [\begin{matrix} d^{1} d^{1} & d^{1} d^{2} & - d^{1} \\ d^{2} d^{1} & d^{2} d^{2} & - d^{2} \\ - d^{1} & - d^{2} & 1 \end{matrix}],

(54)

where $(a^{αβ}) : = {(a_{αβ})}^{- 1}$ and $d^{α} : = a^{αβ} d_{β}$ . Note that formula (54) can be proved by directly computing the product of the two matrices appearing in the right-hand sides of (52) and (54).

Then, the definition of the Frobenius norm combined with the estimates for the coefficients of the matrix field $A^{- 1}$ obtained in part (ii) of the proof implies that, at each point of ω,

\begin{matrix} | G^{- 1} | & = {(Tr (G^{- T} G^{- 1}))}^{1 ∕ 2} = {(Tr (G^{- 1} G^{- T}))}^{1 ∕ 2} = (Tr {(A^{- 1})}^{1 ∕ 2} \\ \leq C δ^{- 11}, \end{matrix}

(55)

for some constant $C = C (ω, p) \in ℝ$ .

Furthermore, the components of the matrix fields $(r_{αβ}) : = {(a_{αβ})}^{1 ∕ 2}$ and $({\tilde{r}}_{αβ}) : = {({\tilde{a}}_{αβ})}^{1 ∕ 2}$ satisfy

r_{αβ} = [a_{11} + a_{22} + 2 {(\det (a_{αβ}))}^{1 ∕ 2}]^{- 1 ∕ 2} {a_{αβ} + {(\det (a_{αβ}))}^{1 ∕ 2} δ_{αβ}}

(56)

and

{\tilde{r}}_{αβ} = [{\tilde{a}}_{11} + {\tilde{a}}_{22} + 2 {(\det ({\tilde{a}}_{αβ}))}^{1 ∕ 2}]^{- 1 ∕ 2} {{\tilde{a}}_{αβ} + {(\det ({\tilde{a}}_{αβ}))}^{1 ∕ 2} δ_{αβ}},

(57)

as consequences of Cayley–Hamilton formula applied to the matrix fields $(a_{αβ}) : ω \to ℝ^{2 \times 2}$ and $({\tilde{a}}_{αβ}) : ω \to ℝ^{2 \times 2}$ .

It follows that the matrix fields defined by (40), viz.,

G : = (\begin{matrix} r_{11} & r_{12} & r_{1 α} a^{αβ} d_{β} \\ r_{21} & r_{22} & r_{2 α} a^{αβ} d_{β} \\ 0 & 0 & {(n - a^{στ} d_{σ} d_{τ})}^{1 ∕ 2} \end{matrix})

and

\tilde{G} : = (\begin{matrix} {\tilde{r}}_{11} & {\tilde{r}}_{12} & {\tilde{r}}_{1 α} {\tilde{a}}^{αβ} {\tilde{d}}_{β} \\ {\tilde{r}}_{21} & {\tilde{r}}_{22} & {\tilde{r}}_{2 α} {\tilde{a}}^{αβ} {\tilde{d}}_{β} \\ 0 & 0 & {(\tilde{n} - {\tilde{a}}^{στ} {\tilde{d}}_{σ} {\tilde{d}}_{τ})}^{1 ∕ 2} \end{matrix})

belong to the space $W^{1, p} (ω; ℝ^{3 \times 3})$ , since their components are products of functions in $W^{1, p} (ω)$ , which is an algebra.

Moreover, these matrix fields satisfy the estimate

\begin{matrix} ∥ \tilde{G} - G ∥_{W^{1, p} (ω)} & \leq \sum_{αβ} ∥ {\tilde{r}}_{αβ} - r_{αβ} ∥_{W^{1, p} (ω)} \\ + \sum_{α} ∥ {\tilde{r}}_{ασ} {\tilde{a}}^{στ} {\tilde{d}}_{τ} - r_{ασ} a^{στ} d_{τ} ∥_{W^{1, p} (ω)} \\ + ∥ {(\tilde{n} - {\tilde{a}}^{αβ} {\tilde{d}}_{α} {\tilde{d}}_{β})}^{1 ∕ 2} - {(n - a^{αβ} d_{α} d_{β})}^{1 ∕ 2} ∥_{W^{1, p} (ω)}, \end{matrix}

on the one hand. On the other hand, we proved previously in parts (ii) and (iii) of the proof that the functions in the right-hand side of the above inequality are bounded above in $W^{1, p} (ω)$ -norm by constants depending only on ω, p, and δ. Then, we can use the triangle inequality in the space $W^{1, p} (ω)$ and the fact that $W^{1, p} (ω)$ is an algebra to estimate all the norms appearing in the right-hand side above. For instance, using the shorter notation $∥ \cdot ∥ : = ∥ \cdot ∥_{W^{1, p} (ω)}$ , we have

\begin{matrix} ∥ {\tilde{r}}_{ασ} {\tilde{a}}^{στ} {\tilde{d}}_{τ} - r_{ασ} a^{στ} d_{τ} ∥ \leq & ∥ ({\tilde{r}}_{ασ} - r_{ασ}) {\tilde{a}}^{στ} {\tilde{d}}_{τ} ∥ + ∥ r_{ασ} ({\tilde{a}}^{στ} - a^{στ}) {\tilde{d}}_{τ} ∥ \\ + ∥ r_{ασ} a^{στ} ({\tilde{d}}_{τ} - d_{τ}) ∥ \\ \leq & ∥ ({\tilde{r}}_{ασ} - r_{ασ}) ∥ ∥ {\tilde{a}}^{στ} ∥ ∥ {\tilde{d}}_{τ} ∥ + ∥ r_{ασ} ∥ ∥ {\tilde{a}}^{στ} - a^{στ} ∥ ∥ {\tilde{d}}_{τ} ∥ \\ + ∥ r_{ασ} ∥ ∥ a^{στ} ∥ ∥ {\tilde{d}}_{τ} - d_{τ} ∥ . \end{matrix}

Then, we estimate the norms $∥ ({\tilde{r}}_{ασ} - r_{ασ}) ∥$ , $∥ {\tilde{a}}^{στ} ∥$ , $∥ r_{ασ} ∥$ , $∥ {\tilde{a}}^{στ} - a^{στ} ∥$ , and $∥ a^{στ} ∥$ appearing in the right-hand side of the above inequality by using the explicit formula (45) for the functions $a^{αβ}$ and ${\tilde{a}}^{αβ}$ and the explicit formulas (56) and (57) for the functions $r_{αβ}$ and ${\tilde{r}}_{αβ}$ , respectively, in terms of the functions $a_{αβ}$ and ${\tilde{a}}_{αβ}$ .

Applying this procedure several times, we find a constant $C (ω, p, δ) \in ℝ$ such that

\begin{matrix} ∥ \tilde{G} - G ∥_{W^{1, p} (ω)} \leq C (ω, p, δ) {\sum_{α, β} ∥ {\tilde{a}}_{αβ} - a_{αβ} ∥_{W^{1, p} (ω)} \\ + \sum_{α} ∥ {\tilde{d}}_{α} - d_{α} ∥_{W^{1, p} (ω)} + ∥ \tilde{n} - n ∥_{W^{1, p} (ω)}} . \end{matrix}

(58)

Now, we can estimate the norm $| (\tilde{F} - R F) (y_{0}) |$ . First, we infer from the definition (41) of the orthogonal matrix field R and from the invariance under rotations of the Frobenius norm of matrices that

\begin{matrix} | \tilde{F} - R F | & = | R^{T} \tilde{F} - F | \\ = | F G^{- 1} {\tilde{G}}^{- T} ({\tilde{F}}^{T} \tilde{F}) - F | \\ = | F G^{- 1} {\tilde{G}}^{- T} ({\tilde{G}}^{T} \tilde{G}) - F | \\ = | F G^{- 1} (\tilde{G} - G) | in ω . \end{matrix}

Then, estimate (55) implies that

| \tilde{F} - R F | \leq C (ω, p, δ) | \tilde{G} - G | in ω,

for some constant $C (ω, p, δ) \in ℝ$ . Consequently, using estimate (58) and the continuous embedding $W^{1, p} (ω) \subset C_{b}^{0} (ω)$ , we deduce that

\begin{matrix} | (\tilde{F} - R F) (y_{0}) | & \leq ∥ \tilde{F} - R F ∥_{L^{\infty} (ω)} \\ \leq C (ω, p, δ) {\sum_{α, β} ∥ {\tilde{a}}_{αβ} - a_{αβ} ∥_{W^{1, p} (ω)} \\ + \sum_{α} ∥ {\tilde{d}}_{α} - d_{α} ∥_{W^{1, p} (ω)} + ∥ \tilde{n} - n ∥_{W^{1, p} (ω)}}, \end{matrix}

(59)

for some other constant $C (ω, p, δ) \in ℝ$ .

It remains to estimate the norm $∥ {\tilde{C}}_{α} - C_{α} ∥_{L^{p} (ω)}$ appearing in the right-hand side of inequality (51). This estimate is obtained in the same way as the estimate (58) for $∥ \tilde{G} - G ∥_{W^{1, p} (ω)}$ , the only difference being that the components $C_{αj}^{i}$ of the matrix field $C_{α}$ , which are defined by relations (17), also depend on the functions $k_{αβ}$ . Thus, without mentioning the details, we find in this way a constant $C (ω, p, δ) \in ℝ$ such that

\begin{matrix} ∥ {\tilde{C}}_{α} - C_{α} ∥_{L^{p} (ω)} & \leq C (ω, p, δ) {\sum_{α, β} ∥ {\tilde{a}}_{αβ} - a_{αβ} ∥_{W^{1, p} (ω)} \\ + \sum_{α} ∥ {\tilde{d}}_{α} - d_{α} ∥_{W^{1, p} (ω)} + ∥ \tilde{n} - n ∥_{W^{1, p} (ω)} \\ + \sum_{α, β} ∥ {\tilde{k}}_{αβ} - k_{αβ} ∥_{L^{p} (ω)}} . \end{matrix}

(60)

In conclusion, using inequalities (59) and (60) in the right-hand side of inequality (51) implies that

\begin{matrix} ∥ \tilde{F} - R_{0} F ∥_{W^{1, p} (ω)} & \leq C (ω, p, δ) {\sum_{α, β} ∥ {\tilde{a}}_{αβ} - a_{αβ} ∥_{W^{1, p} (ω)} \\ + \sum_{α} ∥ {\tilde{d}}_{α} - d_{α} ∥_{W^{1, p} (ω)} + ∥ \tilde{n} - n ∥_{W^{1, p} (ω)} \\ + \sum_{α, β} ∥ {\tilde{k}}_{αβ} - k_{αβ} ∥_{L^{p} (ω)}}, \end{matrix}

(61)

for some constant $C (ω, p, δ) \in ℝ$ .

v. Proof of inequality (39) of the theorem.

The mapping

θ : = \tilde{φ} - (a_{0} + R_{0} φ)

belongs to the space $W^{2, p} (ω; ℝ^{3})$ since the mappings φ and $\tilde{φ}$ belong to this space. In particular, then, $θ \in W^{1, p} (ω; ℝ^{3}) \subset C_{b}^{0} (ω; ℝ^{3})$ by the Sobolev embedding theorem, so its value at $y_{0} \in ω$ is well defined. Besides, the definition (42) of the vector $a_{0}$ implies that

θ (y_{0}) = 0 .

Then, Poincaré–Wirtinger inequality in the space $W^{1, p} (ω; ℝ^{3})$ shows that (see, e.g., Adams and Fournier [25])

∥ θ ∥_{L^{p} (ω)} \leq C_{1} (ω, p) \sum_{α} ∥ \partial_{α} θ ∥_{L^{p} (ω)},

for some constant $C_{1} (ω, p) \in ℝ$ . Consequently,

∥ \tilde{φ} - (a_{0} + R_{0} φ) ∥_{L^{p} (ω)} \leq C_{1} (ω, p) \sum_{α} ∥ \partial_{α} \tilde{φ} - R_{0} \partial_{α} φ ∥_{L^{p} (ω)},

which in turn implies that

∥ \tilde{φ} - (a_{0} + R_{0} φ) ∥_{W^{2, p} (ω)} \leq C_{2} (ω, p) \sum_{α} ∥ \partial_{α} \tilde{φ} - R_{0} \partial_{α} φ ∥_{W^{1, p} (ω)},

for some other constant $C_{2} (ω, p) \in ℝ$ . Therefore,

\begin{matrix} ∥ \tilde{φ} - (a_{0} + R_{0} φ) ∥_{W^{2, p} (ω)} + ∥ \tilde{d} - R_{0} d) ∥_{W^{1, p} (ω)} \\ \leq C_{2} (ω, p) \sum_{α} ∥ \partial_{α} \tilde{φ} - R_{0} \partial_{α} φ ∥_{W^{1, p} (ω)} + ∥ \tilde{d} - R_{0} d) ∥_{W^{1, p} (ω)} . \end{matrix}

The vector fields ${\partial_{α} φ, d}$ are the column vectors of the matrix field F (see part (i) of the proof) and the vector fields ${\partial_{α} \tilde{φ}, \tilde{d}}$ are the column vectors of the matrix field $\tilde{F}$ . Hence, the previous inequality implies that

\begin{matrix} ∥ \tilde{φ} - (a_{0} + R_{0} φ) ∥_{W^{2, p} (ω)} + ∥ \tilde{d} - R_{0} d) ∥_{W^{1, p} (ω)} \\ \leq C_{3} (ω, p) ∥ \tilde{F} - R_{0} F ∥_{W^{1, p} (ω)}, \end{matrix}

for some constant $C_{3} (ω, p) \in ℝ$ .

Inequality (39) of the theorem is then obtained by combining this inequality with the inequality (61) estimating its right-hand side. This completes the proof of the theorem. □

Remark 6. In the classical setting of surface theory, whereby the director fields of any two given surfaces $S : = φ (ω)$ and $\tilde{S} : = \tilde{φ} (ω)$ are respectively defined by

d = a_{3} : = \frac{\partial_{1} φ \land \partial_{2} φ}{| \partial_{1} φ \land \partial_{2} φ |} and \tilde{d} = {\tilde{a}}_{3} : = \frac{\partial_{1} \tilde{φ} \land \partial_{2} \tilde{φ}}{| \partial_{1} \tilde{φ} \land {\tilde{\partial}}_{2} φ |},

we have ${\tilde{k}}_{αβ} = - {\tilde{b}}_{αβ}$ , $k_{αβ} = - b_{αβ}$ , ${\tilde{d}}_{α} = d_{α} = 0$ , and $\tilde{n} = n = 1$ . Then, the inequality (39) established in Theorem 7 becomes

\begin{matrix} ∥ \tilde{φ} - (a_{0} + R_{0} φ) ∥_{W^{2, p} (ω)} + ∥ {\tilde{a}}_{3} - R_{0} a_{3} ∥_{W^{1, p} (ω)} \\ \leq C {\sum_{α, β} ∥ {\tilde{a}}_{αβ} - a_{αβ} ∥_{W^{1, p} (ω)} + \sum_{α, β} ∥ {\tilde{b}}_{αβ} - b_{αβ} ∥_{L^{p} (ω)}}, \end{matrix}

which is precisely the inequality obtained by Ciarlet and Mardare [22, Theorem 5.2].

Footnotes

Acknowledgements

Part of the work of Cristinel Mardare has been completed while he was visiting the Sino-French Institute at Renmin University of China in Suzhou, whose hospitality is hereby gratefully acknowledged.

ORCID iD

Cristinel Mardare

Funding

The work of Maria Malin has been supported by a grant of the Romanian Ministry of Research, Innovation and Digitalization (MCID), project number 22—Nonlinear Differential Systems in Applied Sciences, within PNRR-III-C9-2022-I8.

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Recovery and continuity of surfaces with assigned director field

Abstract

Keywords

1. Introduction

2. Notation and definitions

3. The geometry of a surface with assigned director field

4. Recovery of a surface with assigned director field

5. The Gauss and Codazzi–Mainardi equations

6. Continuity of the surface with assigned director field as a function of its invariants ( a α β , k α β , d α , n )

Footnotes

Acknowledgements

ORCID iD

Funding

References

6. Continuity of the surface with assigned director field as a function of its invariants $(a_{α β}, k_{α β}, d_{α, n})$