Homogenization of diffusion problems with a nonlinear interfacial resistance
type: Item
instance of: scholarly article
title: Homogenization of diffusion problems with a nonlinear interfacial resistance
zbMATH Open document ID: 1339.35022
author: P. Donato
author: Kim Hang Le Nguyen
published in: NoDEA. Nonlinear Differential Equations and Applications
publication date: 2015-10-15T00:00:00Z
review text: The authors study the asymptotic behaviour as \(\varepsilon \to 0\) of the following problems given in an open bounded and with Lipschitz boundary set \(\Omega \subset {\mathbb R}^n\): \[ \begin{cases} - \operatorname{div} ( A^{\varepsilon} (x) \nabla u_1^{\varepsilon} (x) ) + h_1^{\varepsilon}(x, u_1^{\varepsilon}(x)) = f (x) & \text{in } \Omega_1^{\varepsilon} \\ - \operatorname{div} ( A^{\varepsilon} (x) \nabla u_2^{\varepsilon} (x) ) + h_2^{\varepsilon}(x, u_2^{\varepsilon}(x)) = f (x) & \text{in } \Omega_2^{\varepsilon} \\ A^{\varepsilon} (x) \nabla u_1^{\varepsilon} (x) \cdot n_1^{\varepsilon} = - A^{\varepsilon} (x) \nabla u_2^{\varepsilon} (x) \cdot n_2^{\varepsilon} & \text{in } \Gamma^{\varepsilon} \\ - A^{\varepsilon} (x) \nabla u_1^{\varepsilon} (x) \cdot n_1^{\varepsilon} = \varepsilon^{\gamma + 1} (x) h^{\varepsilon} (x, u_1^{\varepsilon} (x) - u_2^{\varepsilon}(x)) & \text{in } \Gamma^{\varepsilon} \\ u_1^{\varepsilon} = 0 & \text{in } \partial \Omega \cap \partial \Omega_1^{\varepsilon} \\ u_2^{\varepsilon} = 0 & \text{in } \partial \Omega \cap \partial \Omega_2^{\varepsilon} \end{cases} \] where \(\Omega_2^{\varepsilon}\) represents a structure of periodic and unvaried holes with Lipschitz boundary and \(\Omega_1^{\varepsilon} = \Omega \setminus \Omega_2^{\varepsilon}\), while \(\Gamma^{\varepsilon}\) represents the boundary of \(\Omega_2^{\varepsilon}\). \(A^{\varepsilon}\) is a matrix defined as \[ A^{\varepsilon} (x) := A \big( x / \varepsilon) \] where \(A\) is a matrix whose entries are functions in \(L^{\infty} (\Omega)\) and satisfies \[ \big( A (x) \lambda, \lambda \big) \geqslant \alpha |\lambda|^2 \, , \qquad \big| A (x) \lambda \big| \leqslant \beta |\lambda| \] for every \(\lambda \in {\mathbb R}^n\) and for two given positive constant \(\alpha\) and \(\beta\). The peculiarity and the difficulty rely on the fact that \(h\), \(h_1\), \(h_2\) may be non-linear functions satisfying suitable assumptions.
reviewed by: Fabio Paronetto
Mathematics Subject Classification ID: 35B27
Mathematics Subject Classification ID: 82B24
Mathematics Subject Classification ID: 35J57
Mathematics Subject Classification ID: 35J61
Mathematics Subject Classification ID: 35J66
zbMATH DE Number: 6494628
zbMATH Keywords: periodic homogenization
zbMATH Keywords: elliptic equations with jumps
zbMATH Keywords: nonlinear interface conditions
MaRDI profile type: Publication
full work available at URL: s00030-015-0325-2
OpenAlex ID: W624192197
cites work: Homogenization and Two-Scale Convergence
cites work: EVOLUTION AND MEMORY EFFECTS IN THE HOMOGENIZATION LIMIT FOR ELECTRICAL CONDUCTION IN BIOLOGICAL TISSUES
cites work: A hierarchy of models for the electrical conduction in biological tissues via two-scale convergence: the nonlinear case
cites work: Macroscopic modelling of heat transfer in composites with interfacial thermal barrier
cites work: Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata]
cites work: Q4189539
cites work: Existence and uniqueness for a quasilinear elliptic problem with nonlinear Robin conditions
cites work: Homogenization of a quasilinear elliptic problem with nonlinear Robin boundary conditions
cites work: Homogénéisation d'un problème de diffusion en milieu composite avec barrière à l'interface
cites work: Conduction of heat in solids
cites work: Periodic unfolding and homogenization
cites work: The Periodic Unfolding Method in Homogenization
cites work: The periodic unfolding method in domains with holes
cites work: Q3836478
cites work: The periodic unfolding method in perforated domains
cites work: Q3619618
cites work: Homogenization in open sets with holes
cites work: Homogenization of reticulated structures
cites work: Periodic homogenization of monotone multivalued operators
cites work: Q3595128
cites work: Q2930359
cites work: Homogenization of the wave equation in composites with imperfect interface: a memory effect
cites work: Correctors for the Homogenization of a Class of Hyperbolic Equations with Imperfect Interfaces
cites work: Corrector results for a parabolic problem with a memory effect
cites work: HOMOGENIZATION OF TWO HEAT CONDUCTORS WITH AN INTERFACIAL CONTACT RESISTANCE
cites work: The periodic unfolding method for a class of imperfect transmission problems
cites work: Q4447127
cites work: Model of diffusion in partially fissured media
cites work: Q3438861
cites work: Homogenization for heat transfer in polycrystals with interfacial resistances
cites work: Q5189325
cites work: Heat Conduction in Fine Scale Mixtures With Interfacial Contact Resistance
cites work: Composites with imperfect interface
cites work: Q4459659
cites work: Q4652779
cites work: A General Convergence Result for a Functional Related to the Theory of Homogenization
cites work: Q4866871
cites work: Upscaling in Nonlinear Thermal Diffusion Problems in Composite Materials
DOI: 10.1007/S00030-015-0325-2
Recommended article: Homogenization in the problems of nonlinear diffusion
Recommended article: Q3068709
Recommended article: The homogenization of diffusion-convection equations in non-periodic structures
Recommended article: Diffusion problems in multi-layer media with nonlinear interface contact resistance
Recommended article: Asymptotic Behaviour of the Interface of Some non-linear Diffusion Problems
Recommended article: Homogenization of nonlinear equations with mixed boundary conditions
Recommended article: Q5189325
Recommended article: Q5371323
Recommended article: Some nonlinear homogenization problems
Recommended article: Q5271587
description: scientific article; zbMATH DE number 6494628