A nonlinear Korn inequality estimates the distance between two immersions from an open subset of into the Euclidean space , , in terms of the distance between specific tensor fields that determine the two immersions up to a rigid motion in . We establish new inequalities of this type in two cases: when k = n, in which case the tensor fields are the square roots of the metric tensor fields induced by the two immersions, and when k = 3 and n = 2, in which case the tensor fields are defined in terms of the fundamental forms induced by the immersions. These inequalities have the property that their constants depend only on the open subset over which the immersions are defined and on three scalar parameters defining the regularity of the immersions, instead of constants depending on one of the immersions, considered as fixed, as up to now.
Nonlinear Korn inequalities are useful in nonlinear three-dimensional elasticity (in which case ) and in nonlinear elastic shell theory (), to estimate the displacement fields in terms of appropriate measures of strain inside an elastic body or shell, such as the Green–St Venant strain tensor in three-dimensional elasticity, respectively the change of metric and of curvature tensor fields in shell theory. They are also useful in differential geometry, e.g., to prove the stability of a surface recovered from given fundamental forms, under the assumptions of the fundamental theorem of surface theory.
In this paper, we establish new nonlinear Korn inequalities that improve previous inequalities of this type in three-dimensional elasticity and in shell theory (see, in particular, the review of these inequalities by Ciarlet et al. [1, 2]) that provide, in addition, explicit estimates of the constants appearing in these inequalities.
A nonlinear Korn inequality for immersions from a domain of into the k-dimensional Euclidean space , , asserts that, given a sufficiently smooth and orientation-preserving immersion and two specific normed vector spaces X and Y, there exists a constant and an exponent such that, for each orientation-preserving immersion such that ,
where and denote specific tensor fields that determine the immersions and , respectively, up to a rigid motion in . Note that this property of the tensor fields appearing in the right-hand side implies that the left-hand side of this inequality vanishes if, and only if, its right-hand side does.
If and with , in which case the appropriate tensor fields and are the square roots of the metric tensor fields induced by the two immersions, then inequality (1) holds with : see Ciarlet and Mardare [3, Theorem 1(a)]. For other possible choices of tensors fields and spaces in inequality (1) in the case see Ciarlet and Mardare [3] and the brief survey by Ciarlet et al. [1].
If and in which case and are specific matrix fields defined in terms of the fundamental forms of the hypersurfaces defined by the two immersions, then inequality (1) holds with : see Malin and Mardare [2, Theorem 3.1]. For other possible choices of tensors and spaces in inequality (1) in the case with , see Blanchard and Griso [4] and Ciarlet et al. [2, 5].
The objective of this paper is to show that the constant can be replaced in inequality (1) by a constant independent of . This objective is achieved in Theorems 3.4 and 4.4. In these theorems, the mapping is restricted to a subset , where denotes the set of all orientation-preserving immersions of class in the case and a similar set in the case , that can be made as large as desired by letting the parameters and go to (in which case converges to the whole set in the sense of set theory for each given parameter ).
Note that it is not possible to let belong to the whole set , as one can infer from the following example due to H. Le Dret (personal communication): the mappings and , defined for each by
are orientation-preserving and share the same metric tensor field, but there is no proper orthogonal matrix such that
The paper comprises four sections, including this introduction. Section 2 specifies the notation and gathers all necessary definitions and preliminary results. Section 3 presents the results in the case where . Section 4 presents the results in the case where with . Note that the choice is made merely for simplicity. Indeed, all the results of Section 4 can be extended to the general case of an arbitrary integer by adopting the methodology used by Malin and Mardare [6, 7] in order to establish nonlinear Korn inequalities for hypersurfaces in .
2. Notation and definitions
This section collects the notation and definitions used throughout this paper.
The notation denotes the n-dimensional Lebesgue measure of the set between the bars.
Given an open subset of , the geodesic distance in between two points is defined by
In other words, the geodesic distance in between two points x and y is the length of the “shortest curve”contained in that joins x and y (the “inf” in the right-hand side ensures that the geodesic distance is well defined even if such a “shortest curve” does not exist).
A domain in , , is a connected and open subset of that is in addition bounded and has a Lipschitz-continuous boundary (in the sense of Adams [8], Maz’ya [9], or Nečas [10]), the domain being then locally on only one side of its boundary.
An important property of domains (as opposed to general connected open subsets) is that the geodesic distance in a domain is equivalent with the Euclidian distance, in the following sense: For each domain, there exists a constantsuch that
where |·| denotes the Euclidean norm; for a proof, see, e.g., Anicic et al. [11, Proposition 5.1].
Another important property of domains is that the space , which is defined as the set of all functions such that f and all its partial derivatives possess continuous extensions to the closure of , coincides with the set of all restrictions to of functions in ; see Ciarlet and Mardare [12] for this specific result, or Whitney [13] for a general theory of extension of differentiable functions.
An immersion from an open subset of into , , is classically a smooth enough mapping whose (classical) partial derivatives, which form a family of n vector fields defined over with values in , are linearly independent at each point of . This definition is naturally extended to mappings in Sobolev spaces, by replacing classical partial derivatives by weak partial derivatives, and to mappings defined up to the boundary of : thus, in what follows, an immersion designates either a mapping whose (classical) partial derivatives are linearly independent at each point of , or a mapping whose weak partial derivatives are linearly independent at almost all points of .
Throughout this paper, vector and matrix fields are denoted by boldface letters to distinguish them from scalar fields (functions). Spaces of vector or matrix fields are denoted by specifying the domain of definition and the target space; for instance, the notation designates the space of all vector fields such that for all .
The gradient of a vector field is the matrix field denoted and defined by
where i denotes the row index and are the Cartesian coordinates of a point in .
Here and in what follows, denotes the space of all matrices with real coefficients. If , we let . The subset of formed by all the proper orthogonal matrices is denoted . The subset of formed by all symmetric matrices is denoted . The subset of formed by all positive-definite matrices is denoted .
The square root of a matrix is the unique matrix such that ; henceforth, the square root of will be denoted . The identity matrix in is denoted (same notation for all ). The notation designates the block matrix formed by two matrices A and B that have the same number of rows. The notation designates the matrix with as its component at the ith line and jth column.
Lebesgue and Sobolev spaces of functions are denoted and , respectively. Lebesgue spaces of vector and matrix fields are respectively denoted and defined by
and Sobolev spaces of vector and matrix fields are respectively denoted and defined by
The spaces and are respectively endowed with the norms denoted and defined by
where |·| denotes the Euclidean norm, and by
where |·| denotes the Frobenius norm. Note that these norms are invariant under rotations, in the sense that if is a orthogonal matrix and is a orthogonal matrix, then
3. Nonlinear Korn inequalities for mappings from into
In this section, we establish two lemmas and two theorems.
The first theorem improves an inequality about the “geometric rigidity” of mappings from into , established in Ciarlet and Mardare [3, Lemma 2], by estimating the constant appearing in this inequality. The second theorem improves a “nonlinear Korn inequality in a domain of ”, established by Ciarlet and Mardare [3, Theorem 1(a)], again by estimating the constant appearing in this inequality.
To begin with, we establish two lemmas which are needed in the proof of Theorem 3.3. They are also interesting in their own right.
Throughout this section, n designates an integer . A current point in is denoted and partial derivative operators with respect to are denoted . Latin indices and exponents such as range in the set .
The set of all orientation-preserving immersions of class from the closure of an open subset of into is denoted
It is clear that, for each mapping defined over a domain , there exists a constant such that
Then the subsets
of the set form an increasing family of sets such that
The following lemma, which is a generalization for of a previous result by Ciarlet and Mardare [14, Lemma 5], will be needed in the proof of Theorem 3.3. Note that the matrix in the next statement in not unique, and that its existence is guaranteed by the compactness of the subset of .
Lemma 3.1.Let there be given an integer , two real numbers and , and a domain .
Given any non-empty open subset U of and any two mappings and , let
and let be any matrix that satisfies
Then the matrixdepends only on the restrictions to the subset U of of the mappings andand
for all non-empty open subsets V of such that .
Proof. Let be the conjugate exponent of p and let be any matrix in . Then
and, thanks to the definitions of and ,
Therefore,
where
Then the conclusion follows by noting that the assumption that belongs to the set implies that
□
Given any mapping , , and any real number , there exists a constant such that
(remember that |·| denotes either the Frobenius or the Euclidean norm of the matrix or vector between the bars).
Then, given any real numbers and , the family of sets
where
have the following two properties:
and
The implicit function theorem (see, e.g., Ciarlet [15]) and the regularity of the boundary of together imply that every mapping is locally injective: for each , there exists an open ball centred at x with radius depending on x such that the restriction of to the set is one-to-one; cf. Ciarlet and Mardare [14].
The following lemma gives (in particular) a uniform lower bound for this “radius of injectivity” when belongs to the subset of the set . Note that the existence of the real number in the following statement is guaranteed by the assumption that is a domain: cf. Section 2.
Lemma 3.2.Given any integer and any domain , let be any real number such that
Then the following two assertions hold.
(a) For each real numberand for each mapping,
for all.
(b) For each real number, and, and for each mapping,
for all such that.
Proof. Given two points x and in , there exists and a mapping such that
and
cf. the definition of the geodesic distance in Section 2.
(i) Let . Then, for each ,
Then the inequality of part (a) of the lemma is obtained by using the above relation, inequality (3) satisfied by , and inequality (2) satisfied by the geodesic distance :
(ii) Let . Then relation (8) implies that, for each such that ,
Since, for each ,
and since belongs to the set , we have
(see inequality (5)).
Using this inequality in the right-hand side of above estimate of implies that
Combined with the inequality and with inequality (2) satisfied by the geodesic distance in , the above inequality next implies that
Since the mapping belongs to the set , it satisfies inequalities (3); consequently, for each , the eigenvalues of the matrix belong to the interval and so
Using this estimate in the right-hand side of the previous inequality implies that
as claimed in the statement of part (b) of the lemma. □
We are now in a position to establish the announced inequality about the “geometric rigidity” of mappings from into . It improves a previous inequality of Ciarlet and Mardare [3, Lemma 2] by estimating the constant appearing in this inequality.
Note that the set appearing in Theorem 3.3, which is the same as that defined by relation (6), can be made as large as desired in the set of all orientation-preserving immersions from into :
for each parameter ; cf. relations (4) and (7).
Note also that the assumption that made in Theorem 3.3 is justified by Lemma 3.2(b), and that it can be replaced by the weaker assumption
in which case the conclusion of Theorem 3.3 holds with a constant depending also on the new parameter .
Theorem 3.3.Given an integer , a domain in , and four real numbers , , and , let
and let be any real number such that
Assume that. Then there exists a constantsuch that
for all mappingsand.
Proof. For clarity, the proof is divided into four steps, numbered (i) to (iv).
(i) Decomposition of into a finite union of well chosen domains.
Let . The assumption that the boundary of is Lipschitz-continuous implies that, for each point , there exists a Cartesian frame centred at , an open neighbourhood of of the form
where and satisfy and , and a Lipschitz-continuous function such that
Note that the restrictions and on the size of the neighbourhood of will be needed later in the proof to apply Lemma 3.2(b). For more details about the regularity of the boundary of open and connected subsets of , see, e.g., Adams [8], Maz’ya [9] or Nečas [10].
Clearly,
(the constants a and b depend in particular on , as already mentioned). Since is bounded, there exists a positive integer and points , , such that
where and .
Let
It is clear from the construction of the above open cover of that there exists a constant such that the set
satisfies
If is the empty set, then relation (11) shows that
where .
If , then we define an open cover for by complementing the open cover (11) of with an open cover of constructed as follows.
Let denote the open n-cube with edge length , centred at the point , and with edges parallel to the axes of coordinates in the Euclidean space . Then the above definition of the number r implies that
Furthermore, since the closure of the set is a compact set such that
there exists an integer and points , , such that
where
Relations (11) and (12) together show that
where and the open sets are defined by relations (10) and (13).
The above decomposition of the set has the following three properties that will be needed in the next steps of the proof.
First, a recursion argument implies that there exists a bijection such that, for each , the set is connected. Therefore, even if it means re-numbering if necessary the sets , we can assume without loss of generality that, for each , the set
is connected.
Second, in the case where , the set is necessarily non-empty for all (because the set is connected and ). Since the set is also an open set in , its n-dimensional Lebesgue measure is strictly positive. Therefore, since the index k varies in a finite set whose cardinal depends only on and , there exists a positive constant such that, for each ,
Third, Lemma 3.2(b) combined with the assumption that and with the construction of the set as being contained in an open ball with diameter implies that, for each mapping and for each , the mapping
is one-to-one. Therefore, the image
by the mapping of the open and connected set is also open and connected (thanks to the invariance of domain theorem; see, e.g., Ciarlet [15]).
Furthermore, Lemma 3.2 implies that, for each , the mapping
is bi-Lipschitz with both Lipschitz constants independent of (by definition, a bi-Lipschitz mapping is a Lipschitz-continuous mapping that possesses an inverse that is also Lipschitz-continuous). Specifically, both the mapping and its inverse mapping are Lipschitz-continuous with a Lipschitz constant given by
(ii) There exists a constant such that, for each real number , for each mappings and , and for each index ,
Pick any mapping . We prove that there exists such that, for each , the set is the image of a n-cube by a bi-Lipschitz mapping with Lipschitz constant (the same for all indices j).
If , then is the image of the n-cube (see relation (13)) by the mapping , which is bi-Lipschitz with Lipschitz constant (see Step (i)).
If , then is the image of the set (see relation (10)) by the mapping , which is bi-Lipschitz with Lipschitz-constant (see Step (i)). We now show that is itself the image of a n-cube by a bi-Lipschitz mapping.
Given any , let denote the n-cube with edge length in and let be the mapping defined by
where
are the vectors of the Cartesian basis appearing in the definition of the boundary of (see Step (i)), and is the Lipschitz-continuous function appearing in the same definition.
Then the mapping is Lipschitz-continuous, one-to-one, onto the set
and its inverse mappings is also Lipschitz-continuous. This last assertion is a simple consequence of the explicit expression of the mapping :
where is defined by
with
(the dot · denotes the Euclidean inner product in ).
Since the index j varies in a finite set with elements, since the functions are Lipschitz-continuous and depend only on and , and since the constants and depend also only on and (see Step (i)), both the mappings and its inverse mapping are Lipschitz-continuous with a same Lipschitz constant, denoted in what follows by
Therefore,
which shows that is the image of the n-cube by the composite mapping , which is clearly bi-Lipschitz with Lipschitz constant .
We just proved that, for each , the set is the image of a n-cube by a bi-Lipschitz mapping with Lipschitz constant
This allows us to apply the geometric rigidity lemma for mappings from into due to Friesecke et al. [16] (see also Conti [17] for extensions to ). Accordingly, there exists a constant such that
for all and for all . That the constant can be chosen independently of j follows from the fact that it can be chosen uniformly for a family of domains which are images of n-cubes by bi-Lipschitz mappings with a same Lipschitz constant, and from the fact that this constant is also invariant under dilations (so the edge-length of the cube is irrelevant): cf. Friesecke et al. [18, Theorem 5].
Now, pick any mapping . Thanks to the definition of the set in Step (i) and to the properties of the mapping established previously, the mapping
belongs to the space . Consequently, the previous inequality implies that
then that
Noting that, at each ,
and
we infer from the previous inequality that the inequality announced in Step (ii) holds with
Since depends only an p and and since itself depends only on and , the constant depends only on and .
(iii) Let the constant and integer be those defined in Step (i). Then there exists a constant such that, for each mapping , for each positive real number , and for each mapping ,
Let there be given a positive real number and two mappings and .
If the decomposition found in Step (i) holds with , then the above inequality holds with as proven in Step (ii).
Thus, assume that . Remember that for all ; cf. Step (i).
Since , the inequality established in Step (ii) implies that
with .
Assume that, for some , there exists a constant such that
As shown in Step (ii), one also has
Since the set is non-empty and is contained in both and , Lemma 3.1 implies that, for the same matrix defined in terms of the restrictions of and to the set (see Lemma 3.1), we have
and
Since , the last four inequalities together with the definition of the constant in step (i) imply that
Thus, a recursion argument shows that the inequality announced in Step (iii) holds with a constant defined by the recurrence relation: and
for .
(iv) Conclusion
The parameters and J appearing previously depend only on and (cf. Step (i)) and the constant depends only on and (cf. Step (ii)). Therefore, the constant found previously depends only on and , so the inequality established in Step (iii) satisfies the conclusion of the theorem. □
We conclude this section by establishing a new nonlinear Korn inequality in a domain , with a constant depending only on and on three scalar parameters p, and .
Note that the constant appearing in the following statement is the same as the constant appearing in Theorem 3.3, and so its dependence on the parameters and is given explicitly in the proof of Theorem 3.3.
Theorem 3.4.Given an integer , a domain in and four real numbers , , and , let
let
and let be any real number such that
Assume that. Then there exists a constant such that
for all mappingsand.
Proof. It is well known that the mapping
is one-to-one and onto, and that the inverse mapping
Let and let . Since the determinant of the matrix is then positive, the symmetric matrix is positive-definite, which, in turn, implies that the matrix is well-defined and belongs to the set . Since, in addition, the matrix field is continuous, the matrix field
is also continuous.
Let . Since the determinant of the matrix is then positive for almost all , the symmetric matrices are positive-definite for almost all , which in turn implies that the matrices are well-defined and belong to the set for almost all . Since, in addition, the matrix field is measurable, the mapping
is also measurable.
Define the matrices
and
Then
for all , and
for almost all . It follows that the matrices and belong to for almost all , then that belongs to for almost all .
Then the invariance under rotations of the Frobenius norm and the above polar decompositions of the matrix fields and imply that, on the one hand,
for almost all .
On the other hand, since , we can apply Theorem 3.3 to the mappings and and deduce in this way that
where the constant depends only on and on the given parameters .
Then the conclusion follows by combining the last two inequalities. □
4. Nonlinear Korn inequalities for mappings from into
In this section, we establish two lemmas and two theorems.
The first theorem improves an inequality about the “geometric rigidity” of immersions from into due to Malin and Mardare [6, Lemma 3.2], and the second theorem improves a “nonlinear Korn inequality on a surface” due to Malin and Mardare [6, Theorem 3.1] by specifying the constants involved.
To begin with, we establish two lemmas which are needed in the proof of first theorem in this section (Theorem 3.3). They are also interesting in their own right in view of their possible applications in differential geometry.
Note that the mappings appearing in this section are defined from a two-dimensional set into the three-dimensional Euclidean space, by contrast to the previous section, where we considered mappings from a n-dimensional set into the Euclidean space with the same dimension n.
Note also that the results of this section can be extended to mappings from a n-dimensional set, with an arbitrary integer, into the Euclidean space with dimension , by adopting the methodology introduced in Malin and Mardare [6, 7], where they established several nonlinear Korn inequalities for hypersurfaces in (instead of surfaces in here).
Throughout this section, Latin indices and exponents such as range in the set , while Greek indices and exponents such as range in the set . The summation convention for repeated indices or exponents is used in conjunction with these rules (for instance, the relation appearing in the following is short notation for ).
Let be an open subset of , let denote the Cartesian coordinates of a generic point in , and let denote the corresponding partial derivative operators.
A mapping is an immersion if
If is an immersion, we let
If is an immersion such that , we let
and
Remark.(a) If the immersion is of class , then the image
is locally a surface in in the classical sense, the vector field
is the positively oriented unit normal vector field to the surface S, the functions
are the covariant and contravariant components, respectively, of the first fundamental form of the surfaceS, and the functions
are the covariant and mixed components, respectively, of the second fundamental form of the surfaceS.
(b) The matrix fieldis the inverse of the matrix field.
Note that the matrix field is , symmetric and positive-definite almost everywhere in , so that its square root is well defined almost everywhere in . Since is itself a positive-definite symmetric matrix, its inverse is also well-defined.
The following lemma is straightforward generalization of Lemmas 3.2 and 3.3 of Ciarlet et al. [5] and for this reason its proof is omitted. This shows that some classical definitions and properties pertaining to surfaces in still hold under less-stringent regularity assumptions than is usual, which are traditionally given and established under the assumptions that the surface is the image of an immersion of class from into .
Lemma 4.1.Given any domain and any real number , let
and
where is the vector field defined by (14).
Given any bounded open intervaland any mappingthat belongs either to the set, or to the set, define the mappingby letting
Then the following assertions hold.
(a) If, then
and
(b) If, then
and
Given any domain in and any mapping , there exists a real number such that
Given in addition any constant , there exists a real number such that
where |·| denotes either the Euclidean norm or the Frobenius norm of the vector or matrix between the bars.
It follows from these observations that, for each real number ,
where
Note that this decomposition of the set is in fact a limit of an sequence of increasing sets:
hence, the smaller and , the larger .
The following lemma establishes a key property of mappings that belong to the set . Note that the existence of the real number in the following statement is guaranteed by the assumption that is a domain: cf. Section 2.
Lemma 4.2.Given any domain and any real numbers , and , let
let
and let the subsetsandbe defined by (9) and (18), respectively.
Given any mapping, letand let the mappingbe defined by
Then
Proof. Throughout this proof, we use the short-hand notation , , and (see (14) and (15) for the definitions of these vector and scalar fields), and we let
for all . Note that and are the mean and total curvature of the surface at the point , respectively.
The definition of the mapping implies that its gradient at is the block matrix
Consequently, at each ,
and
Since the mapping belongs to the set , which means that it satisfies, in particular, inequalities (16), the previous inequality implies that, for each ,
Combined with Weingarten equations
the above expression of the determinant of implies that, for each ,
Hence, we have, on the one hand,
for all .
On the other hand, we infer from the definition (15) of the functions and that
and
Since the mapping belongs to the set , we have, for all ,
Using the last three inequalities in the previous estimate of the determinant of yields the desired lower bound for this determinant:
for all .
It remains to estimate for all such that .
Since and for some and , we deduce from the definition of that
which next implies that
The last term on the right-hand side of the previous inequality, namely , can be estimated in terms of as follows. Let . Then
where
In order to estimate the right-hand side in the previous expression of , we first deduce from the assumption that the mapping belongs to the set that, at each ,
and
Therefore,
Besides,
so that
Combined with the relations (recall that the functions are defined by relations (15)) and with the inequality
the previous inequality implies that
Using this inequality on the right-hand side of the previous estimate of , we finally obtain
We are now in a position to estimate the right-hand side of inequality (22). Using the previous inequality, we first deduce that
Since implies that , we infer from the definition of the space and from the definition of the scalars and in the statement of the lemma that
The conclusion follows by noting that estimates (20), (21) and (23) together imply that the mapping belongs to the set ; cf. definition (9) of this set. □
We are now in a position to establish the announced inequality about the “geometric rigidity” of mappings from into . It improves a previous inequality of Malin and Mardare [6, Lemma 3.2] (see also [7, Lemma 1]) by estimating the constant appearing in this inequality.
Note that the set appearing in Theorem 4.3 can be made as large as desired be choosing sufficiently small parameters and : indeed,
for all parameters ; cf. relation (19).
Note also that the assumption that made in Theorem 4.3 is derived from the assumption made in Theorem 3.3 on the parameter . The existence of the constant in the statement of the Theorem below is guaranteed by the assumption that is a domain: cf. Section 2. Finally, the vector field appearing in the definition of the set is defined by (cf. relation (14))
Theorem 4.3.Given a domain in and any real numbers , , and , let
letbe the subset ofdefined by (18), and let be any real number such that
Assume that. Then there exists a constantsuch that
for all mappingsand.
Proof. Given parameters and as in the statement of the theorem, let , , ,
and let the sets be defined by (9).
Given any mapping , define the mapping by letting
Then Lemma 4.2 implies that
Given any mapping , define the mapping by letting
Then Lemma 4.1 implies that
We apply Theorem 3.3 to the mappings and defined previously. To this end, we have to prove that and that
That follows from the assumption combined with the previous definitions of the parameters and .
Let and be any two points in the set . Given any mapping defining a curve joining y and in , the mapping defines a curve that joins the point x to in , which is a subset of . In addition, the point is joined to the point by a segment of length contained in . Then we infer from the definition of the geodesic distances in and in (Section 2) that
for all points and in .
Combined with the definition of the constant in the statement of the theorem, the previous inequality implies the desired result:
Since the real numbers and the mappings and satisfy all the assumptions of Theorem 3.3, we have
for some constant depending only on , p, and (see the proof of Theorem 3.3 for an explicit dependence), which themselves depend only on , p, and .
The remainder of the proof consists of finding a lower bound, respectively an upper bound, for the left-hand side, respectively right-hand side, of inequality (24).
Thanks to the definition of the mappings and and of the Frobenius norm, we first have
where
Thanks to the symmetry of the domain with respect to the variable and to Clarkson’s inequalities in the space with (see, e.g., Adams [8, Theorem 2.28]), we next have
Using that the vector fields and are independent of , we deduce by combining the last two inequalities that
Consequently, since and ,
We next find a lower bound for the right-hand side of the previous inequality by using the inequalities
(which clearly follows from the definitions of the vector fields and and of the Frobenius norm), and the inequality
(which is a consequence of Jensen’s inequality for a concave function).
Combining the last four inequalities, we finally obtain the following lower bound for the left-hand side of inequality (24):
where
is a constant that depends only on and p (since ).
It remains to find an upper bound for the right-hand side of inequality (24) in terms of the mappings and and the corresponding vector fields and .
Let denote any matrix in the set .
First, for all and for almost all , we have
Second, using either Jensen’s inequality if , or the inequality for all non-negative real numbers if , we deduce from the above inequality that
with a constant depending only on p, given explicitly by
Using once again Jensen’s inequality in the last term on the right-hand side of the previous inequality yields
Finally, by integrating the previous inequality in from to , we obtain
on the one hand.
On the other hand, Fubini’s theorem and the properties of the infimum imply that
The last two inequalities together imply that
Now we estimate the right-hand side of the previous inequality by using inequalities and , then the inequalities
and
and, finally, the inequality for a specific choice of non-negative real numbers (which holds since ). In this fashion, we deduce that
hence, that
Combining the previous inequality with inequalities (24) and (25) obtained previously implies the inequality announced in the statement of the theorem with a constant given by
Note that the constant depends only on the set and on the parameters . Since , and , the constant depends, in fact, only on the set and on the parameters , as claimed in the statement of the theorem. □
We conclude this section by establishing a new nonlinear Korn inequality on a surface , with a constant that depends on only via two scalar parameters and .
Note that the constant appearing in the statement of Theorem 4.4 is given explicitly by
in terms of the constant of Theorem 4.3; see the end of the proof of Theorem 4.4. Hence, its dependence on the parameters and is given explicitly in the proof of Theorem 4.3.
We use the following notation: Given any mapping that belongs either to the set , or to the set , we let as before (see relations (14) and (15))
and
and we define the matrix fields
from into . Remember that is the positively oriented unit normal vector field to the surface , that the functions and are the covariant components of the first and second fundamental forms of S, respectively, and that the matrix field is positive definite, so it possesses a unique square root; cf. Section 2.
Theorem 4.4.Given a domain in and any real numbers , , and , let
let be the subset ofdefined by (18), and letbe any real number such that
Assume that. Then there exists a constantsuch that
for all mappingsand.
Proof. Let two mappings and be given once and for all.
The assumption that allows us to apply Theorem 4.3 to these functions and obtain the inequality
for some constant depending only on and . Then the properties of the infimum and Cauchy–Schwarz inequality in further imply that
where the function is defined by
for almost all .
Define the matrices
for almost all , and the matrices
for all . Then
for all matrices and for almost all , so that
for almost all .
Define next the matrix fields
and
Then for almost all and for all , so that
for almost all . Then the previous inequality implies that
for almost all .
Define the matrix fields
and observe that the following relations hold:
and
where the notation 0 denotes the zero vector in , namely .
Define next the matrix fields
and note that the equation of Weingarten satisfied by the partial derivatives of the vector fields and , namely,
imply that
and
Using the previous expressions of the matrix fields , , , and , as well as the invariance under rotations of the Frobenius norm, in the previous estimate of yields the following inequality:
for almost all .
The definition of , , implies that the function is measurable and Lemma 4.1 implies that
By integrating the previous inequality over , we then deduce that
Combining this inequality with inequality (26) obtained at the beginning of the proof shows that the inequality announced in Theorem 4.4 holds with
□
Footnotes
Acknowledgements
The first author would like to thank Professor Philippe Ciarlet for his invitation to City University of Hong Kong, where this manuscript has been partially completed.
Funding
The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The work of the first author was supported by a grant from the Romanian Ministry of Research and Innovation, CNCS-UEFISCDI (project number PN-III-P1-1.1-TE-2019-0397), within PNCDI III. The work of the second author was substantially supported by a grant from City University of Hong Kong (project number 7200659).
ORCID iD
Cristinel Mardare
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