Spectral transition and variational localization in multilayer transport with interface resistance and dissipation

Abstract

We investigate transport in multilayer systems combining heterogeneous diffusion, interfacial resistance, and volumetric dissipation. Unlike classical diffusion models, the presence of dissipative bulk terms and imperfect interfaces leads to a dissipative diffusion–transmission operator structure. We show that the spectral properties of the associated diffusion–efflux operator are governed by a single mechanism: the competition between volumetric dissipation and interfacial transmission. This competition is quantified by a dimensionless parameter that controls both the principal eigenvalue and the spatial structure of the corresponding eigenfunction. In particular, the system exhibits a transition between extended and localized modes, providing a spectral mechanism for transport limitation in heterogeneous media. Variational localization arises as a variational consequence of dissipative contrasts and does not require modification of the constitutive diffusion law. The formulation admits a natural interpretation within the framework of generalized continuum mechanics, where interface terms define a surface energy analogous to micromorphic or Cosserat-type interactions. Numerical experiments confirm the predicted scaling laws and the emergence of variational localization across regimes. These results provide a unified spectral framework for understanding transport in systems with internal interfaces and dissipation.

Keywords

multilayer diffusion spectral operators Eigenfunction localization generalized continuum mechanics interface transmissions

1. Introduction

Transport in heterogeneous media is often governed by the combined effects of diffusion, interfacial transmission, and internal dissipation. Such situations arise naturally in composite materials, porous media, and biological systems, where transport occurs across multiple layers with distinct properties and imperfect interfaces [1 –5]. In many of these systems, the presence of internal interfaces and volumetric removal mechanisms leads to transport behavior that cannot be captured by classical homogeneous diffusion models.

From a mathematical perspective, these processes give rise to elliptic operators combining three distinct contributions: bulk diffusion within each layer, transmission conditions across interfaces, and reactive or dissipative terms in the volume. Diffusion in composite media and transmission problems have been extensively studied in the context of homogenization theory and elliptic partial differential equations [1,3,5 –7]. Likewise, elliptic operators with discontinuous coefficients and interface conditions have been analyzed in connection with layered structures and composite materials [8 –10]. Reactive terms have also been investigated in diffusion–reaction equations and dissipative systems [11,12].

However, when volumetric dissipation and interfacial resistance are present simultaneously, the resulting operator exhibits a fundamentally different structure. In particular, the system is no longer conservative: mass is removed through volumetric sink terms, and transport across interfaces is impeded by finite transmission. As a consequence, the long-time behavior of the system is governed not only by diffusion but also by a competition between bulk dissipation and interfacial coupling. Despite the extensive literature on each of these effects taken separately, their combined influence on the spectral properties of the associated operator remains insufficiently understood.

The objective of this work is to provide a rigorous analysis of this combined effect through the spectral structure of the diffusion–efflux operator. We focus on the principal eigenvalue and its associated eigenfunction, which determine the dominant decay rate and spatial structure of the system [13,14]. Our goal is to identify the mechanism governing transport across layered domains and to characterize how it depends on the relative strength of volumetric and interfacial contributions.

Our analysis reveals that the behavior of the system is governed by a single mechanism: the competition between volumetric dissipation and interfacial transmission. This competition is quantified by a dimensionless parameter

Π = \frac{kL}{P}

which compares the strength of bulk dissipation to interface coupling. Here:

k denotes an effective volumetric dissipation (or efflux) coefficient,

P is an effective interface permeability coefficient,

L is a characteristic length scale of the multilayer structure.

We show that this parameter controls both the magnitude of the principal eigenvalue and the spatial structure of the corresponding eigenfunction.

In particular, the system exhibits a transition between two qualitatively distinct regimes. When interfacial coupling dominates, the principal mode extends across the entire domain, and the multilayer structure behaves effectively as a single medium. In contrast, when volumetric dissipation dominates, the principal mode localizes in regions of weaker absorption, leading to transport limitation. This variational localization¹ phenomenon arises as a direct consequence of the variational structure of the operator and is closely related to eigenfunction concentration in heterogeneous media [15,16].

A key aspect of the present work is the interpretation of the model within the framework of generalized continuum mechanics. The interfacial terms define a surface energy penalizing discontinuities across internal boundaries, which can be interpreted as microstructural interactions analogous to micromorphic coupling or Cosserat-type interactions. From this perspective, the spectral problem characterizes the dominant mode of a dissipative continuum with interfacial structure, providing a direct link between operator theory and the mechanics of heterogeneous media.

The analysis is based on a variational formulation of the problem, in which the operator is associated with a bilinear form combining bulk and interfacial contributions. This approach allows us to derive spectral bounds, identify scaling regimes, and obtain quantitative variational localization estimates for the principal eigenfunction. The theoretical predictions are supported by numerical experiments, which confirm the scaling behavior of the principal eigenvalue and the emergence of variational localization across regimes.

The novelty of the present work is not the introduction of a new diffusion–reaction model itself, but the identification of a spectral transition mechanism generated by the combined action of volumetric dissipation and imperfect interfacial transmission. While diffusion–reaction transmission problems have been studied extensively in homogenization and elliptic interface theory, the present work shows that the principal eigenpair of the associated operator exhibits two distinct regimes governed by the dimensionless ratio Π. In particular, the analysis reveals: (1) spectral scaling laws linking the dominant eigenvalue to bulk and interface dissipation, (2) variational localization of the principal mode in weakly dissipative regions, and (3) a variational interpretation of this variational localization mechanism in layered heterogeneous media. The contribution is therefore primarily spectral and variational rather than constitutive.

Although the motivation for this study is rooted in transport phenomena across biological membranes—where interface permeability and active removal play a key role [17 –21]—the framework developed here applies more broadly to diffusion processes in heterogeneous systems with imperfect interfaces and dissipation.

The paper is organized as follows. Section 2 introduces the biophysical foundations of the model and discusses the underlying transport mechanisms. Section 3 presents the multilayer formulation and its variational structure, including the non-equilibrium nature of the operator. Section 4 is devoted to the spectral analysis, where the main result establishing the transition between regimes is stated. Section 5 provides a detailed interpretation of this transition, including its connection to generalized continuum mechanics. Section 6 develops quantitative scaling laws and variational localization estimates. Finally, Section 7 presents numerical experiments validating the theoretical predictions.

2. Biophysical motivation

Transport across bacterial envelopes provides a prototypical example of diffusion in heterogeneous media with interfaces and active dissipation. In particular, Gram-negative bacteria exhibit a multilayered cell-envelope architecture in which fundamentally different transport mechanisms coexist and interact. A schematic cross-section of this architecture is shown in Figure 1, and the individual components are described in detail below.

Figure 1.

Cross-sectional schematic of the Gram-negative bacterial cell envelope and its multilayer transport mechanisms.

2.1. Envelope architecture

The Gram-negative cell envelope consists of three principal compartments separated by two membrane barriers (Figure 1):

Outer membrane (OM). The outermost layer is an asymmetric lipid bilayer whose outer leaflet is composed predominantly of lipopolysaccharide (LPS) and whose inner leaflet consists of phospholipids [17]. This composition confers a strong permeability barrier, severely limiting the passive diffusion of hydrophilic and amphiphilic compounds. Transport across the OM is therefore not spatially homogeneous: it occurs primarily through discrete proteinaceous channels known as porins (e.g. OmpF, OmpC in Escherichia coli; OprD in Pseudomonas aeruginosa), which create a spatially heterogeneous effective permeability [17,21]. Lipophilic compounds additionally experience significant resistance arising from the ordered LPS matrix.

Periplasmic space (PP). Sandwiched between the two membranes is the periplasm, an aqueous gel-like compartment containing the peptidoglycan (murein) cell wall. The periplasm acts as an intermediate reservoir: solutes that have crossed the OM equilibrate within this space before encountering the inner membrane. Its composition and electrostatic properties can further modulate transport kinetics [18].

Inner (cytoplasmic) membrane (IM) and cytoplasm. The IM is a symmetric phospholipid bilayer that constitutes the primary osmotic and energetic boundary of the cell. Unlike the OM, it does not contain large aqueous channels; passive permeation is therefore restricted to small, uncharged, or highly lipophilic molecules. The cytoplasm enclosed by the IM is the site of metabolism and gene expression.

2.2. Active efflux and multidrug resistance

A key feature distinguishing Gram-negative transport from simple passive diffusion is the presence of active efflux systems. Multidrug resistance (MDR) efflux pumps of the resistance-nodulation-division (RND) family continuously expel a broad spectrum of compounds—including antibiotics, detergents, and metabolic by-products—from the cytoplasm directly to the extracellular space [19,20].

These tripartite complexes span all three envelope compartments:

the inner-membrane transporter (e.g., MexB in P. aeruginosa, AcrB in E. coli) harnesses the proton-motive force to drive substrate translocation from the cytoplasm into the periplasm;

the periplasmic adaptor protein (MexA, AcrA) bridges the two membrane components and stabilizes the functional complex;

the outer-membrane channel (OprM, TolC) provides a continuous conduit from the periplasm to the extracellular medium, bypassing re-entry into the cytoplasm.

The best-characterized examples are MexAB–OprM in Pseudomonas aeruginosa [19] and AcrAB–TolC in Escherichia coli [20], both of which are clinically significant determinants of intrinsic and acquired antibiotic resistance. By continuously removing substrate from the interior, these pumps introduce a non-conservative component to the transport process: the total intracellular mass is not conserved but decreases at a rate proportional to the local concentration.

2.3. Competition between influx and efflux

The interplay between passive influx through porins and active efflux through pump complexes governs the steady-state intracellular concentration of any permeating compound. When efflux is strong relative to the permeability of the OM, solutes are expelled faster than they enter, leading to a pronounced depletion of the intracellular concentration:

\begin{matrix} C_{in} ≪ C_{ext} . \end{matrix}

(1)

This inequality is a hallmark of intrinsic antibiotic resistance in Gram-negative pathogens [18,19] and motivates the central modeling assumption of this work: the dynamics are governed by the competition between influx across imperfect interfaces and volumetric removal in the bulk.

From a modeling perspective, this competition leads naturally to a description in terms of diffusion in a composite medium with transmission conditions at interfaces and reactive (dissipative) terms in the bulk. Specifically, the interface permeability coefficient $P_{ij}$ encodes the effective conductance of the OM porin channels, while the volumetric dissipation coefficient $k_{i}$ represents the net rate of active removal by efflux pumps. The relative strength of these two mechanisms is quantified by the dimensionless parameter

\begin{matrix} Π = \frac{k L}{P}, \end{matrix}

(2)

where L is a characteristic length scale of the multilayer structure. As shown in Sections 4 to 7, Π acts as a spectral control parameter: for $Π ≪ 1$ the dominant transport mode extends across the entire multilayer structure (permeability-dominated regime), whereas for $Π ≫ 1$ the principal eigenfunction undergoes variational localization near the OM (efflux-dominated regime), providing a spectral mechanism for the concentration depletion expressed in (1).

This setting motivates the study of diffusion operators incorporating heterogeneous coefficients, interfacial resistance, and dissipation, which are analyzed rigorously in the following sections.

3. Continuum model

3.1. Geometric setting

Let $Ω \subset ℝ^{d}$ be a bounded domain representing a multilayer medium. We assume that Ω admits a partition into a finite number of subdomains

Ω = ⋃_{i = 1}^{N} Ω_{i},

where each $Ω_{i}$ is an open set with Lipschitz boundary. The interfaces between adjacent layers are denoted by

Γ_{ij} = \partial Ω_{i} \cap \partial Ω_{j},

for $i \neq j$ .

3.2. Governing equations

Let $c_{i} : Ω_{i} \times (0, T) \to ℝ$ denote the concentration field in layer $Ω_{i}$ . In each subdomain, the evolution is governed by a diffusion–reaction equation of the form

\begin{matrix} \partial_{t} c_{i} = \nabla \cdot (D_{i} \nabla c_{i}) - k_{i} c_{i}, x \in Ω_{i}, t > 0, \end{matrix}

(3)

where $D_{i}$ is a symmetric positive definite diffusion tensor and $k_{i} \geq 0$ is a reaction coefficient.

3.3. Interface conditions

Across each interface $Γ_{ij}$ , we impose transmission conditions that couple the solutions in adjacent layers. These consist of flux continuity

\begin{matrix} D_{i} \nabla c_{i} \cdot n_{ij} = D_{j} \nabla c_{j} \cdot n_{ij}, \end{matrix}

(4)

together with an imperfect transmission condition

\begin{matrix} - D_{i} \nabla c_{i} \cdot n_{ij} = P_{ij} (c_{i} - c_{j}), \end{matrix}

(5)

where $P_{ij} \geq 0$ is an interface permeability coefficient and $n_{ij}$ denotes the unit normal vector on $Γ_{ij}$ pointing outward from $Ω_{i}$ .

3.4. Boundary and initial conditions

The system is supplemented with suitable boundary conditions on $\partial Ω$ (e.g., Dirichlet or Neumann conditions) and initial data

c_{i} (x, 0) = c_{i}^{0} (x), x \in Ω_{i} .

3.5. Variational formulation

We introduce the Hilbert space

V = \prod_{i = 1}^{N} H^{1} (Ω_{i}),

equipped with the natural norm

∥ u ∥_{V}^{2} = \sum_{i = 1}^{N} ∥ u_{i} ∥_{H^{1} (Ω_{i})}^{2} .

The weak formulation of the problem is obtained by multiplying each equation by a test function $v_{i} \in H^{1} (Ω_{i})$ and integrating by parts. This leads to the bilinear form $a : V \times V \to ℝ$ defined by

\begin{matrix} a (u, v) & = \sum_{i = 1}^{N} \int_{Ω_{i}} D_{i} \nabla u_{i} \cdot \nabla v_{i} dx + \sum_{i = 1}^{N} \int_{Ω_{i}} k_{i} u_{i} v_{i} dx \\ + \sum_{(i, j)} \int_{Γ_{ij}} P_{ij} (u_{i} - u_{j}) (v_{i} - v_{j}) dS . \end{matrix}

(6)

The corresponding weak problem reads: find $c \in L^{2} (0, T; V)$ such that, for all $v \in V$ ,

\begin{matrix} \frac{d}{dt} \sum_{i = 1}^{N} \int_{Ω_{i}} c_{i} v_{i} dx + a (c, v) = 0, \end{matrix}

(7)

in a suitable weak sense.

3.6. Operator formulation

The bilinear form $a (\cdot, \cdot)$ defines an elliptic operator $L$ on $L^{2} (Ω)$ with domain determined by the transmission conditions. The operator takes the form

L c = - \nabla \cdot (D \nabla c) + kc,

with interface contributions incorporated through the variational formulation.

This operator will serve as the basis for the spectral analysis developed in the subsequent sections.

3.7. Non-equilibrium nature of the transport operator

The transport dynamics considered here differ fundamentally from classical diffusion due to the combined presence of heterogeneous diffusion, interfacial transmission, and volumetric dissipation. These mechanisms introduce a non-conservative structure in the governing operator, beyond standard Fickian diffusion.

A key consequence is the lack of mass conservation. Indeed, the total mass satisfies

\begin{matrix} \frac{d}{dt} \int_{Ω} c (x, t) dx = - \int_{Ω} k (x) c (x, t) dx \leq 0, \end{matrix}

(8)

reflecting the presence of volumetric sink terms. The system therefore exhibits intrinsically dissipative dynamics.

From an operator-theoretic perspective, this non-equilibrium character² is encoded in the spectral structure of the diffusion–efflux operator. In particular, the competition between interfacial resistance and volumetric dissipation governs both the principal eigenvalue and the spatial structure of the associated eigenfunctions.

As a result, the dominant modes may exhibit strong spatial heterogeneity, including variational localization in subregions of weak dissipation. This behavior does not arise from modifications of the constitutive diffusion law, but directly from the variational structure of the operator.

This observation provides the foundation for the spectral transition described in Theorem 1.

4. Spectral structure and regime transition

We now analyze the spectral properties of the multilayer diffusion–dissipation operator introduced in Section 3. The goal is to identify the mechanisms governing the principal eigenvalue and the spatial structure of the corresponding eigenfunction and to characterize the transition between transport-dominated and dissipation-dominated regimes.

4.1. Operator setting and variational structure

Let $Ω = ⋃_{i = 1}^{N} Ω_{i}$ be a partitioned domain with interfaces $Γ_{ij} = \partial Ω_{i} \cap \partial Ω_{j}$ . The Hilbert space is

V = \prod_{i = 1}^{N} H^{1} (Ω_{i}),

endowed with the natural norm.

The bilinear form $a : V \times V \to ℝ$ is given by

a (u, v) = \sum_{i = 1}^{N} \int_{Ω_{i}} D_{i} \nabla u_{i} \cdot \nabla v_{i} dx + \sum_{i = 1}^{N} \int_{Ω_{i}} k_{i} u_{i} v_{i} dx + \sum_{(i, j)} \int_{Γ_{ij}} P_{ij} (u_{i} - u_{j}) (v_{i} - v_{j}) dS .

Under standard assumptions on $D_{i}$ , $k_{i}$ , and $P_{ij}$ , the form $a (\cdot, \cdot)$ is symmetric, continuous, and coercive. It therefore defines a self-adjoint operator L on $L^{2} (Ω)$ with compact resolvent.

The spectral problem reads: find $(λ, ϕ) \in ℝ \times V$ , $ϕ \neq 0$ , such that

a (ϕ, v) = λ \int_{Ω} ϕv dx for all v \in V .

The principal eigenvalue $λ_{1}$ admits the variational characterization

\begin{matrix} λ_{1} = \inf_{u \in V, u \neq 0} \frac{a (u, u)}{∥ u ∥_{L^{2} (Ω)}^{2}}, \end{matrix}

(9)

and is associated with a nonnegative eigenfunction $ϕ_{1}$ .

4.2. Main spectral result: transition and variational localization

The bilinear form $a (u, u)$ contains two competing dissipative contributions:

\underset{volumetric dissipation}{\underset{⏟}{\sum_{i = 1}^{N} \int_{Ω_{i}} k_{i} u_{i}^{2} dx}} and \underset{interfacial dissipation}{\underset{⏟}{\sum_{(i, j)} \int_{Γ_{ij}} P_{ij} {(u_{i} - u_{j})}^{2} dS}} .

Their interplay governs both the magnitude of $λ_{1}$ and the spatial structure of $ϕ_{1}$ . This is made precise by the following result.

Theorem 1. (Spectral bounds and variational concentration mechanism). Let $(λ_{1}, ϕ_{1})$ be the principal eigenpair associated with the bilinear form $a (\cdot, \cdot)$ , normalized by

∥ ϕ_{1} ∥_{L^{2} (Ω)} = 1 .

Assume that the diffusion tensors are uniformly elliptic and that the boundary conditions and/or the volumetric dissipation eliminate the constant null mode, so that a Poincaré-type inequality holds on the partitioned domain.

Assume further that the effective parameters $k_{eff}$ and $P_{eff}$ are representative scales satisfying

c_{k} k_{eff} \leq k_{i} \leq C_{k} k_{eff}, c_{P} P_{eff} \leq P_{ij} \leq C_{P} P_{eff},

for positive constants independent of $k_{eff}$ , $P_{eff}$ , and L.

Then, there exist positive constants $C_{1}, C_{2}$ , depending only on the geometry, diffusion coefficients, trace inequalities, and comparability constants, such that

\begin{matrix} C_{1} \min {k_{eff}, \frac{P_{eff}}{L}} \leq λ_{1} \leq C_{2} \max {k_{eff}, \frac{P_{eff}}{L}} . \end{matrix}

(10)

Moreover, assume that a region $Ω_{inner} \subset Ω$ satisfies

k (x) \geq κ in Ω_{inner},

with

κ ≫ \frac{P_{eff}}{L} .

Assume also that there exists an admissible comparison function concentrated in a weakly dissipative region outside $Ω_{inner}$ whose Rayleigh quotient is bounded by

\frac{a (w, w)}{∥ w ∥_{L^{2} (Ω)}^{2}} \leq C \frac{P_{eff}}{L} .

Then, the principal eigenfunction satisfies the variational concentration estimate

\begin{matrix} \int_{Ω_{inner}} ϕ_{1}^{2} dx \leq C_{3} \frac{P_{eff}}{κL}, \end{matrix}

(11)

for some constant $C_{3} > 0$ depending only on the geometry and diffusion coefficients.

In particular, the principal eigenvalue is controlled by the competing volumetric and interfacial dissipation scales, while the principal eigenfunction has small $L^{2}$ -mass in regions where the volumetric dissipation is large compared with the interfacial transmission scale.

The proof of this theorem is provided in Appendix 1.

4.3. Spectral interpretation, parametric dependence, and regime structure

Theorem 1 identifies a single dimensionless parameter governing the spectral behavior:

\begin{matrix} Π = \frac{k_{eff} L}{P_{eff}} . \end{matrix}

(12)

The parameter $Π$ is not introduced as a new constitutive quantity, but rather as an emergent spectral control parameter arising from the balance between volumetric dissipation and interface transmission. Its significance lies in the fact that the principal eigenvalue and the spatial structure of the dominant eigenfunction organize into regimes governed by $Π$ , leading to a unified interpretation of transport limitation.

More precisely, $Π$ compares the strength of volumetric dissipation to interface transmission and separates two distinct spectral regimes:

$Π ≪ 1$ : transport-dominated regime, in which interfacial coupling is strong and the principal mode extends across the multilayer structure;

$Π ≫ 1$ : dissipation-dominated regime, in which volumetric removal suppresses transport and the principal mode exhibits variational concentration in regions of weaker dissipation.

This transition is not imposed at the level of the constitutive equations, but emerges from the spectral structure of the operator and the associated variational problem.

The variational characterization (9) also immediately yields monotonicity properties of the principal eigenvalue with respect to the dissipation and transmission coefficients.

Proposition 1. (Monotonicity). The principal eigenvalue $λ_{1}$ is nondecreasing with respect to the coefficients $k_{i}$ and $P_{ij}$ . That is, if ${\tilde{k}}_{i} \geq k_{i}$ and ${\tilde{P}}_{ij} \geq P_{ij}$ , then

{\tilde{λ}}_{1} \geq λ_{1} .

Proof. The result follows directly from the inequality

ã (u, u) \geq a (u, u) for all u \in V,

combined with the variational characterization (9). □

The above results show that the spectral properties of the operator are governed by the competition between volumetric and interfacial dissipation. In particular,

the principal eigenvalue reflects the dominant dissipation mechanism;

the associated eigenfunction encodes the spatial structure of transport;

variational concentration arises as a consequence of heterogeneous dissipation.

The resulting framework provides a unified spectral interpretation of transport limitation in multilayer systems and forms the basis for the quantitative scaling laws and numerical validations developed in the subsequent sections.

5. Interpretation and regime analysis

The spectral results established in Section 4 provide a unified framework for understanding transport in multilayer systems with interface resistance and volumetric dissipation. In particular, the principal eigenpair $(λ_{1}, ϕ_{1})$ encodes both the dominant decay rate and the spatial structure of transport.

5.1. Competing dissipation mechanisms

The variational structure

a (u, u) = \sum_{i = 1}^{N} \int_{Ω_{i}} D_{i} | \nabla u_{i} |^{2} dx + \sum_{i = 1}^{N} \int_{Ω_{i}} k_{i} u_{i}^{2} dx + \sum_{(i, j)} \int_{Γ_{ij}} P_{ij} {(u_{i} - u_{j})}^{2} dS

reveals the presence of two competing dissipative effects: volumetric dissipation within each layer and interfacial dissipation across layer boundaries.

The principal eigenfunction $ϕ_{1}$ minimizes the associated Rayleigh quotient and therefore balances these two contributions. Regions with large $k_{i}$ penalize the presence of the solution, while large $P_{ij}$ penalize discontinuities across interfaces. The observed transport behavior results from this competition.

5.2. Dimensionless control parameter

The spectral behavior identified in Section 4 is governed by the dimensionless parameter

\begin{matrix} Π = \frac{k_{eff} L}{P_{eff}}, \end{matrix}

(13)

which measures the relative strength of volumetric dissipation and interfacial transmission.

Rather than introducing new regimes, this parameter provides a compact interpretation of the spectral transition described in Theorem 1. In particular, Π compares two competing energetic contributions: the bulk penalty associated with volumetric dissipation and the interfacial penalty associated with discontinuities across layers.

From a physical standpoint, Π can be viewed as a ratio of characteristic time scales. Small values correspond to rapid interfacial equilibration compared to bulk dissipation, while large values indicate that volumetric removal dominates over transport across interfaces.

The transition between extended and localized modes therefore emerges as a direct consequence of the relative magnitude of these two mechanisms, as quantified by Π.

5.3. Mechanics interpretation

To emphasize the variational structure independently of the transport interpretation, we denote the generic state variable by u. In the present diffusion setting, u corresponds to the concentration field, rather than to a mechanical displacement. The notation is adopted solely to facilitate comparison with generalized continuum theories.

The variational structure of the bilinear form

a (u, u) = \sum_{i = 1}^{N} \int_{Ω_{i}} D_{i} | \nabla u_{i} |^{2} dx + \sum_{i = 1}^{N} \int_{Ω_{i}} k_{i} u_{i}^{2} dx + \sum_{(i, j)} \int_{Γ_{ij}} P_{ij} {(u_{i} - u_{j})}^{2} dS

admits a natural interpretation within the framework of generalized continuum mechanics with interfacial interactions.

Energy structure and interfacial mechanics. Introducing the jump

{〚 u 〛}_{ij} = u_{i} - u_{j},

the total energy can be written as

E (u) = \sum_{i = 1}^{N} \int_{Ω_{i}} (\frac{1}{2} D_{i} | \nabla u_{i} |^{2} + \frac{1}{2} k_{i} u_{i}^{2}) dx + \sum_{(i, j)} \int_{Γ_{ij}} \frac{1}{2} P_{ij} {〚 u 〛}_{ij}^{2} dS .

The first term represents bulk diffusion and volumetric dissipation, while the second term defines a surface energy penalizing discontinuities across interfaces. The coefficient $P_{ij}$ plays the role of an interfacial stiffness, controlling the degree of continuity enforced between adjacent layers.

In a rate-dependent interpretation, this interface contribution corresponds to a dissipation potential

D_{Γ} (u) = \sum_{(i, j)} \int_{Γ_{ij}} \frac{P_{ij}}{2} {〚 u 〛}_{ij}^{2} dS,

with associated interfacial flux

q_{ij} \sim P_{ij} (u_{i} - u_{j}),

consistent with transmission laws in heterogeneous media.

Micromorphic-type analogy. The multilayer structure can be interpreted as a discrete micromorphic system. Each subdomain $Ω_{i}$ carries its own field $u_{i}$ , and the interfacial jump ${〚 u 〛}_{ij}$ measures the mismatch between neighboring phases.

This is directly analogous to micromorphic models, in which a macroscopic field u is coupled to a micro-field φ through an energy of the form

\int_{Ω} \frac{1}{2} κ | u - φ |^{2} dx .

In the present setting, the role of the mismatch $u - φ$ is played by ${〚 u 〛}_{ij}$ , and the parameter $P_{ij}$ acts as the coupling coefficient. The interfaces therefore encode microstructural interactions in a discrete manner.

Cosserat-type interaction structure. The interface term also admits an interpretation as a continuum analogue of Cosserat-type interaction energies. In discrete generalized continua, one often considers interaction energies of the form

\sum_{(i, j)} K_{ij} | u_{i} - u_{j} |^{2} .

The contribution

\int_{Γ_{ij}} P_{ij} {(u_{i} - u_{j})}^{2} dS

can be viewed as the continuous counterpart of such interactions, where the interfaces act as coupling surfaces between neighboring substructures. The parameter $P_{ij}$ controls the strength of this interaction and determines whether the system behaves as a strongly coupled medium or as weakly interacting layers.

Implications for spectral behavior. Within this generalized continuum framework, the spectral problem associated with $a (\cdot, \cdot)$ characterizes the dominant mode of a system with competing bulk and interfacial energies.

The variational localization phenomenon identified in Theorem 1 admits a direct mechanical interpretation: the system minimizes its energy by concentrating in regions where volumetric dissipation is minimal, while the interfacial energy regulates the degree of coupling between layers.

In the limit of large $P_{ij}$ , the interface penalty enforces approximate continuity and the system behaves as an effective homogeneous medium. Conversely, for small $P_{ij}$ , the coupling weakens and the system exhibits variational localization and decoupling effects.

This interpretation places the diffusion–efflux operator within the class of generalized continua with interfacial interactions and provides a direct link between spectral properties and the mechanics of heterogeneous media.

6. Spectral scaling and variational localization

We now refine the spectral characterization of the operator by quantifying the scaling of the principal eigenvalue and the variational localization properties of the corresponding eigenfunction.

Let $(λ_{1}, ϕ_{1})$ denote the principal eigenpair associated with the bilinear form $a (\cdot, \cdot)$ , normalized by $∥ ϕ_{1} ∥_{L^{2} (Ω)} = 1$ .

6.1. Spectral bounds

The principal eigenvalue is controlled by the competition between volumetric and interfacial dissipation.

Theorem 2. (Spectral bounds). There exist constants $C_{1}, C_{2} > 0$ , depending only on the geometry and ellipticity constants, such that

\begin{matrix} C_{1} \min (k_{eff}, \frac{P_{eff}}{L}) \leq λ_{1} \leq C_{2} \max (k_{eff}, \frac{P_{eff}}{L}) . \end{matrix}

(14)

Proof. The upper bound follows by testing the Rayleigh quotient with constant functions and with functions localized in a single layer. The lower bound follows from coercivity of the bilinear form combined with Poincaré and trace inequalities. □

6.2. Asymptotic regimes

The bounds above imply the following asymptotic behavior.

Proposition 2. (Asymptotic scaling). The principal eigenvalue satisfies

\begin{matrix} λ_{1} & = k_{eff} (1 + o (1)), as Π \to \infty, \end{matrix}

(15)

\begin{matrix} λ_{1} & = \frac{P_{eff}}{L} (1 + o (1)), as Π \to 0 . \end{matrix}

(16)

Proof. The result follows from dominance of the corresponding terms in the Rayleigh quotient in each regime. □

6.3. Variational localization of the principal mode

The spatial structure of the principal eigenfunction reflects the same competition.

Theorem 3. (Variational localization estimate). Let $Ω_{m}$ be a subdomain where the dissipation is minimal, and assume that in a region $Ω_{inner}$ one has

k_{inner} \geq κ ≫ \frac{P_{eff}}{L} .

Then,

\begin{matrix} \int_{Ω_{inner}} ϕ_{1}^{2} dx \leq C \frac{P_{eff}}{κL}, \end{matrix}

(17)

where C depends only on the geometry and diffusion coefficients.

Proof. The estimate follows from the Rayleigh quotient and the upper bound on $λ_{1}$ , which controls the contribution of strongly dissipative regions. □

The above results show that both the magnitude of the principal eigenvalue and the spatial structure of the eigenfunction are governed by the same mechanism: the competition between volumetric and interfacial dissipation.

In particular, the system exhibits a transition from extended modes to localized modes, and the principal eigenvalue reflects the dominant dissipation mechanism. This provides a quantitative spectral characterization of transport limitation in multilayer systems.

7. Numerical validation

7.1. One-dimensional multilayer configuration

We consider a one-dimensional domain $(0, L)$ partitioned into two subdomains,

(0, L) = (0, a) \cup (a, L),

representing an outer and an inner layer, respectively. The concentration fields $c_{1}$ and $c_{2}$ satisfy

\begin{matrix} \partial_{t} c_{1} & = D_{1} c_{1}^{″} - k_{1} c_{1}, x \in (0, a), \end{matrix}

(18)

\begin{matrix} \partial_{t} c_{2} & = D_{2} c_{2}^{″} - k_{2} c_{2}, x \in (a, L), \end{matrix}

(19)

together with transmission conditions at the interface $x = a$ ,

\begin{matrix} D_{1} c_{1}^{'} (a) & = D_{2} c_{2}^{'} (a), \end{matrix}

(20)

\begin{matrix} - D_{1} c_{1}^{'} (a) & = P (c_{1} (a) - c_{2} (a)) . \end{matrix}

(21)

Boundary conditions are prescribed as

\begin{matrix} c_{1} (0, t) & = c_{ext}, \end{matrix}

(22)

\begin{matrix} c_{2}^{'} (L, t) & = 0, \end{matrix}

(23)

and the initial condition is taken as

c (x, 0) = 0 .

7.2. Numerical discretization

The system is discretized on a uniform grid with N nodes and mesh size $Δ x = L ∕ N$ . Second-order central differences are used to approximate the diffusion operator,

c^{″} (x_{i}) \approx \frac{c_{i + 1} - 2 c_{i} + c_{i - 1}}{Δ x^{2}} .

The interface conditions are incorporated by modifying the discrete operator at the interface node. In particular, the permeability condition yields

- D_{1} \frac{c_{a} - c_{a - 1}}{Δ x} = P (c_{a} - c_{a + 1}),

which replaces the standard stencil and enforces coupling between the two subdomains.

Time integration is performed using an implicit Euler scheme,

\frac{c^{n + 1} - c^{n}}{Δ t} = L c^{n + 1},

resulting in a linear system solved at each time step.

7.3. Parameter regime

Unless otherwise specified, we consider the parameter set

L = 1, a = 0.5, D_{1} = 1, D_{2} = 0.1, k_{1} = 0, k_{2} = k,

with permeability parameter $P \in [1 0^{- 3}, 1 0^{2}]$ .

The behavior of the system is characterized by the dimensionless parameter

Π = \frac{kL}{P},

which quantifies the relative strength of volumetric dissipation and interfacial transport.

7.4. Numerical regime analysis and spectral validation

Steady-state solutions are computed for varying values of the permeability parameter P and the volumetric dissipation coefficient k. The resulting concentration profiles exhibit a transition from nearly continuous behavior across the interface at large P to strongly discontinuous profiles at small P, reflecting the increasing influence of interfacial resistance. Increasing k produces a pronounced attenuation of the solution in the inner region, consistent with dominant volumetric dissipation.

To quantify transport into the inner layer, we introduce the integrated concentration

C_{inner} = \int_{a}^{L} c (x) dx .

Numerical results show that $C_{inner}$ decreases monotonically with increasing k, indicating progressive suppression of transport by volumetric removal mechanisms. The transition from significant penetration to effective exclusion occurs over a relatively narrow range of parameters, suggesting a sharp regime change governed by the dimensionless parameter

Π = \frac{kL}{P} .

The dependence of $C_{inner}$ on the permeability parameter P further confirms this interpretation. Increasing P enhances transmission across the interface and increases the concentration within the inner region. This dependence is strongly nonlinear, with the highest sensitivity occurring near the transition regime $Π \approx 1$ , where small variations in permeability produce large changes in transport behavior.

To visualize the global structure of the system, a regime diagram in the $(P, k)$ -plane is constructed by evaluating $C_{inner}$ over a range of parameters. The contour structure reveals two distinct regions associated with $Π ≪ 1$ and $Π ≫ 1$ . The curve $Π = 1$ accurately predicts the transition between these regimes, in agreement with the theoretical spectral analysis.

The spectral structure of the operator is further examined through numerical computation of the principal eigenvalue $λ_{1}$ of the discrete diffusion–transmission operator. The results show that $λ_{1}$ increases with increasing k and decreases with increasing P. When represented as a function of Π, the numerical data exhibit a consistent scaling behavior, supporting the theoretical bounds established in Section 5. In particular, the simulations confirm that the dominant decay rate is controlled by the competition between volumetric dissipation and interfacial transmission.

Overall, the numerical simulations validate the main theoretical predictions of the paper:

the system exhibits two distinct spectral regimes governed by Π;

interface resistance induces concentration discontinuities and delayed transport;

volumetric dissipation strongly suppresses concentration in inner regions;

the principal eigenvalue and eigenfunction accurately capture the observed transport behavior.

These results support the interpretation of transport limitation as a spectral consequence of the balance between permeability and volumetric dissipation in heterogeneous multilayer systems.

8. Results and discussion

8.1. Steady profiles and interface effects

Figure 2 shows steady-state concentration profiles for representative values of the interface permeability P and efflux rate k.

For large permeability ( $P ≫ 1$ ), the solution is nearly continuous across the interface, indicating strong coupling between layers. In contrast, for small P, a jump discontinuity develops, reflecting the dominant contribution of the interfacial term in the transmission condition

- D \nabla c \cdot n = P (c_{i} - c_{j}) .

This behavior is consistent with the variational structure, where small P penalizes flux and allows concentration mismatch across the interface.

Increasing the efflux parameter k induces a strong decay of the solution in the inner region. In the regime of large k, the solution is effectively suppressed beyond the interface, indicating that volumetric dissipation dominates transport.

These observations illustrate the competition between interfacial and volumetric contributions in the energy functional.

8.2. Dependence on efflux

Figure 3 displays the inner concentration

C_{inner} = \int_{a}^{L} c (x) dx

as a function of k for several values of P. The dependence is monotone, with a rapid decrease as k increases.

For fixed P, the transition from significant penetration to negligible concentration occurs over a narrow range of k, indicating a sharp change in the dominant transport mechanism. Larger values of P shift this transition to higher values of k, reflecting the increased ability of the system to transmit mass across the interface.

When expressed in terms of the dimensionless parameter

Π = \frac{kL}{P},

the curves exhibit a consistent scaling behavior, supporting the theoretical prediction that Π governs the transition between regimes.

8.3. Spectral regimes, variational concentration, and transport behavior

Figure 4 shows the inner concentration as a function of the permeability parameter P for different values of the volumetric dissipation coefficient k. As expected, increasing P enhances transmission across the interface and increases the concentration within the inner region. The dependence is strongly nonlinear, with maximal sensitivity occurring near the transition regime $Π \approx 1$ . In this regime, small variations in permeability produce large changes in $C_{inner}$ , indicating competition between interfacial transmission and volumetric dissipation.

To further characterize this transition, Figure 5 presents a contour plot of the inner concentration in the $(P, k)$ plane. The curve

Π = 1

separates two distinct transport regimes:

$Π ≪ 1$ : transport-dominated regime with significant penetration across the multilayer structure;

$Π ≫ 1$ : dissipation-dominated regime with strongly suppressed inner concentration.

The transition between these regimes is sharp and agrees closely with the scaling predicted by the spectral analysis, confirming that Π acts as the relevant spectral control parameter for the system.

The spectral structure of the operator is further illustrated in Figure 6, which shows the principal eigenvalue $λ_{1}$ as a function of Π. The eigenvalue increases with increasing Π, indicating that the dominant decay rate is controlled by the relative strength of volumetric dissipation compared to interfacial transmission. This behavior is consistent with the asymptotic representation

c (x, t) \sim e^{- λ_{1} t} ϕ_{1} (x),

which shows that the long-time dynamics are governed by the principal spectral mode of the operator.

The dependence of $λ_{1}$ on Π therefore establishes a direct quantitative connection between the spectral scaling of the operator and the observed transport regimes.

Figure 7 further demonstrates the change in the spatial structure of the principal eigenfunction across regimes. For $Π ≪ 1$ , the eigenfunction extends across both layers, reflecting strong coupling and efficient transport through the multilayer system. In contrast, for $Π ≫ 1$ , the eigenfunction exhibits variational concentration in the outer region and decays rapidly within the inner layer.

This concentration mechanism follows directly from the variational characterization

λ_{1} = \inf_{u \neq 0} \frac{a (u, u)}{∥ u ∥^{2}},

since regions with large dissipation coefficients contribute strongly to the energy functional. Consequently, minimizing configurations preferentially concentrate in regions where volumetric dissipation is weaker.

Overall, the numerical results show that the observed transport behavior is a direct consequence of the spectral and variational structure of the diffusion–transmission operator. Interface resistance induces concentration discontinuities and limits flux across layers, while volumetric dissipation suppresses transport within strongly absorbing regions. The interplay of these mechanisms produces regime-dependent behavior governed by the dimensionless parameter Π and reflected both in the magnitude of the solution and in the spatial structure of the principal eigenfunction.

These simulations therefore provide numerical validation of the spectral mechanism underlying transport limitation in heterogeneous multilayer systems.

9. Conclusion

We have developed a spectral framework for transport in multilayer systems combining heterogeneous diffusion, interfacial resistance, and volumetric dissipation. The analysis shows that the behavior of the system is governed by a single mechanism: the competition between bulk dissipation and interfacial transmission.

This competition is captured by a dimensionless parameter, which determines both the magnitude of the principal eigenvalue and the spatial structure of the corresponding eigenfunction. In particular, the system exhibits a transition between extended and localized modes, reflecting a change in the dominant dissipation mechanism. This transition arises directly from the variational structure of the operator and does not require modification of the underlying constitutive law.

The variational structure admits an analogy with generalized continuum theories featuring interfacial or microstructural interaction energies. The interface terms define a surface energy penalizing discontinuities, which can be interpreted as microstructural interactions analogous to micromorphic coupling or Cosserat-type interactions. From this perspective, the spectral problem characterizes the dominant mode of a dissipative continuum with interfacial structure.

The theoretical predictions are supported by numerical experiments, which confirm the scaling of the principal eigenvalue and the variational localization of the dominant mode across regimes. Together, these results provide a unified explanation of transport limitation in heterogeneous systems.

The framework developed here opens several directions for future work, including extensions to nonlinear transport, time-dependent interface properties, and the integration of data-driven or operator-learning approaches for complex multiscale systems.

Footnotes

10. Figures

We present here the numerical results supporting the theoretical analysis. The figures illustrate the role of interface permeability, efflux, and spectral structure in shaping transport across the multilayer system.

Appendix 1 ORCID iD

Koffi Enakoutsa

Funding

The author received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Notes

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