Homogenization of a boundary-value problem with a nonlinear boundary condition in a thick junction of type 3:2:1

We consider a boundary-value problem for the Poisson equation in a thick junction Ω_ε, which is the union of a domain Ω₀ and a large number of ε-periodically situated thin curvilinear cylinders. The following nonlinear Robin boundary condition ∂_νu_ε + εκ(u_ε)=0 is given on the lateral surfaces of the thin cylinders. The asymptotic analysis of this problem is performed as ε → 0, i.e. when the number of the thin cylinders infinitely increases and their thickness tends to zero. We prove the convergence theorem and show that the nonlinear Robin boundary condition is transformed (as ε → 0) in the blow-up term of the corresponding ordinary differential equation in the region that is filled up by the thin cylinders in the limit passage. The convergence of the energy integral is proved as well. Using the method of matched asymptotic expansions, the approximation for the solution is constructed and the corresponding asymptotic error estimate in the Sobolev space H¹(Ω_ε) is proved. Copyright © 2007 John Wiley & Sons, Ltd.     

Homogenization of a boundary value problem in a thick cascade junction

We consider a homogenization problem in a singularly perturbed two-dimensional domain of a new type that consists of a junction body and many alternating thin rods of two classes. One of the classes consists of rods of finite length, whereas the other contains rods of small length, and inhomogeneous Fourier boundary conditions (the third type boundary conditions) with perturbed coefficients are imposed on boundaries of thin rods. Homogenization theorems are proved. Bibliography: 38 titles. Illustrations: 2 figures. __________ Translated from Problemy Matematicheskogo Analiza, No. 37, 2008, pp. 47–72.     

Gelfand type elliptic problems under Steklov boundary conditions

For a Gelfand type semilinear elliptic equation we extend some known results for the Dirichlet problem to the Steklov problem. This extension requires some new tools, such as non-optimal Hardy inequalities, and discovers some new phenomena, in particular a different behavior of the branch of solutions and three kinds of blow-up for large solutions in critical growth equations. We also show that small values of the boundary parameter play against strong growth of the nonlinear source.     

Homogenization of a parabolic signorini boundary value problem in a thick plane junction

We consider a parabolic Signorini boundary value problem in a thick plane junction Ω _ε which is the union of a domain Ω₀ and a large number of ε−periodically situated thin rods. The Signorini conditions are given on the vertical sides of the thin rods. The asymptotic analysis of this problem is done as ε → 0, i.e., when the number of the thin rods infinitely increases and their thickness tends to zero. With the help of the integral identity method we prove a convergence theorem and show that the Signorini conditions are transformed (as ε → 0) in differential inequalities in the region that is filled up by the thin rods in the limit passage. Bibliography: 31 titles. Illustrations: 1 figure.     

Homogenization of elliptic problems withL p boundary data

We consider the homogenization problem $$\begin{gathered} - \frac{\partial }{{\partial x_i }}\left( {a^{ij} \left( {\frac{x}{\varepsilon }} \right)\frac{{\partial u_\varepsilon }}{{\partial x_j }}} \right) = 0inD, \hfill \\ u_\varepsilon = gon\partial D, \hfill \\ \end{gathered} $$ whereD is a bounded domain,a is aC ^1,α, periodic, uniformly positive matrix, and the datag belongs toL ^p (?D), 1 <p < ∞. We show that, if?D satisfies a uniform exterior sphere condition, thenu _ε converges in L ^p (D) to the solution of the corresponding homogenized problem asε → 0. The proof is done via estimates for Poisson's kernel. We give examples showing that this convergence result does not hold for a generalG-convergent sequence of operators and depends on the periodicity ofa as well as on its smoothness.     

Homogenization of elliptic boundary value problems in Lipschitz domains

In this paper we study the L ^p boundary value problems for \({\mathcal{L}(u)=0}\) in \({\mathbb{R}^{d+1}_+}\) , where \({\mathcal{L}=-{\rm div} (A\nabla )}\) is a second order elliptic operator with real and symmetric coefficients. Assume that A is periodic in x _d+1 and satisfies some minimal smoothness condition in the x _d+1 variable, we show that the L ^p Neumann and regularity problems are uniquely solvable for 1 < p < 2 + δ. We also present a new proof of Dahlberg’s theorem on the L ^p Dirichlet problem for 2 ? δ < p < ∞ (Dahlberg’s original unpublished proof is given in the Appendix). As the periodic and smoothness conditions are imposed only on the x _d+1 variable, these results extend directly from \({\mathbb{R}^{d+1}_+}\) to regions above Lipschitz graphs. Consequently, by localization techniques, we obtain uniform L ^p estimates for the Dirichlet, Neumann and regularity problems on bounded Lipschitz domains for a family of second order elliptic operators arising in the theory of homogenization. The results on the Neumann and regularity problems are new even for smooth domains.     

Homogenization of elliptic eigenvalue problems: Part 2

The aim of this paper is to study the homogenization of elliptic eigenvalue problems, with a second order homogeneous Dirichlet problem as an example. The main homogenization theorem states that the same operator which serves to homogenize the corresponding static problem works for the eigenvalue problem as well and that the structure of eigenvalues and eigenvectors is in some sense preserved. Formulae for first and second order correctors for eigenvalues are proposed and error estimates are obtained. These results are applied to the case of coefficients with a periodic structure and a simple numerical example is presented. Extensions to other types of boundary conditions and to higher order equations are indicated.

---

Resume Le but de cet article est d'étudier l'homogénéisation du problème de valeurs propres pour des opérateurs elliptiques. On prend comme exemple un problème de second-ordre avec des conditions de Dirichlet homogènes au bord. Le Théorème principal d'homogénéisation dit que le même opérateur qui homogénéise le problème stationnaire correspondant sert également à homogénéiser ce problème de valeurs propres et que la structure des valeurs et vecteurs propres est, grosso modo, préservée. On propose des formules pour calculer les correcteurs de premier et second ordre pour les valeurs propres et on obtient des estimations d'erreur. Ces résultats sont appliqués à un cas particulier où les coefficients sònt périodiques et des résultats numériques sont présentés. On indique des extensions possibles du point de vue conditions aux limites, et des problèmes de quatrième ordre.

---

---

Part 1 of Dr. Kesavan's article appeared in Appl. Math. Optim. 5, Number 2.     

On elliptic problems with indefinite superlinear boundary conditions

In this paper we prove the existence of positive solutions to some nondivergent elliptic equations with indefinite nonlinear boundary conditions. The proof is based on a new Liouville-type theorem about the nonnegative solutions to some canonical indefinite elliptic equations, which is also proved in this paper by the method of moving planes.     

Semilinear elliptic problems with mixed Dirichlet-Neumann boundary conditions

This work deals with the analysis of problems

     

On operators generated by elliptic boundary problems with a spectral parameter in boundary conditions

General elliptic boundary value problems with the spectral parameter appearing linearly both in the elliptic equation and in boundary conditions are considered. It is proved that the corresponding matrix operator from the Boutet de Monvel algebra is similar to an almost diagonal operator. This result is applied to prove the completeness and the summability (in the sense of Abel) of the root vectors of this operator.The support of the Rashi Foundation is gratefully acknowledged.The support of the Israel Ministry of Science and Technology is gratefully acknowledged.     

Homogenization of boundary value problems for monotone operators in perforated domains with rapidly oscillating boundary conditions of fourier type

We deal with homogenization problem for nonlinear elliptic and parabolic equations in a periodically perforated domain, a nonlinear Fourier boundary conditions being imposed on the perforation border. Under the assumptions that the studied differential equation satisfies monotonicity and 2-growth conditions and that the coefficient of the boundary operator is centered at each level set of unknown function, we show that the problem under consideration admits homogenization and derive the effective model. Bibliography: 24 titles.     

Optimal bounds in semilinear elliptic problems with nonlinear boundary conditions

A method (developed by L. E. Payne) of constructing optimal sub- or supersolutions in semilinear elliptic problems is extended to the case of nonlinear boundary conditions. One thus obtains bounds for the solution (or related quantities) which are sharp in the limit as the domain degenerates into a infinite slab.     

Landesman–Lazer Type Conditions and Multiplicity Results for Nonlinear Elliptic Problems with Neumann Boundary Values

We establish the existence and multiplicity of solutions for Steklov problems under non- resonance or resonance conditions using variational methods. In our main theorems, we consider a weighted eigenvalue problem of Steklov type.     

On averaging of nonlinear elliptic problems with inhomogeneous boundary conditions

A sequence of solutions of nonlinear elliptic problems is considered in the case where the Dirichlet conditions are given on the one part of the boundary and the Neumann conditions are given on the other part. The boundary-value problem is constructed.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 2, pp. 269–276, February, 1995.     

椭圆外区域各向异性问题基于人工边界条件的Schwarz交替算法

研究一类二维各向异性外问题的重叠型区域分解.基于自然边界归化,对各向异性外问题提出了一种Schwarz交替算法,并给出其离散形式,分析了算法的收敛性.给出数值试验以示算法的可行性与有效性.     

Some results for non-linear elliptic problems with mixed boundary conditions

We prove existence and uniqueness results for non-linear elliptic equations with lower order terms, L¹ data, and mixed boundary conditions that include as particular cases the Dirichlet and the Neumann problems. Mathematics Subject Classification (2000) 35J25, 35D05, 35J70, 35J60     

Hele-Shaw type problems with dynamical boundary conditions

In this paper, we study the nonlinear evolution equation of Hele-Shaw type with dynamical boundary conditions. That is, the equation ut=Δw+f where u∈H(w) and H is the Heaviside function, with boundary condition μ(x,w)t_∂w+k∇w⋅ν=g, where ν denotes the outward normal vector of the fixed boundary of the domain. We prove existence, uniqueness and some qualitative properties of the solution.