Homogenization of highly oscillating boundaries and reduction of dimension for a monotone problem

Dominique Blanchard; Antonio Gaudiello

Displaying similar documents to “Homogenization of highly oscillating boundaries and reduction of dimension for a monotone problem”

The method of Rothe and two-scale convergence in nonlinear problems

Jiří Vala (2003)

Applications of Mathematics

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Modelling of macroscopic behaviour of materials, consisting of several layers or components, cannot avoid their microstructural properties. This article demonstrates how the method of Rothe, described in the book of K. Rektorys The Method of Discretization in Time, together with the two-scale homogenization technique can be applied to the existence and convergence analysis of some strongly nonlinear time-dependent problems of this type.

Γ-convergence approach to variational problems in perforated domains with Fourier boundary conditions

Valeria Chiadò Piat, Andrey Piatnitski (2010)

ESAIM: Control, Optimisation and Calculus of Variations

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The work focuses on the Γ-convergence problem and the convergence of minimizers for a functional defined in a periodic perforated medium and combining the bulk (volume distributed) energy and the surface energy distributed on the perforation boundary. It is assumed that the mean value of surface energy at each level set of test function is equal to zero. Under natural coercivity and -growth assumptions on the bulk energy, and the assumption that the surface energy satisfies -growth...

Concentration phenomena for solutions of superlinear elliptic problems

Riccardo Molle, Donato Passaseo (2006)

Annales de l'I.H.P. Analyse non linéaire

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Morse index and blow-up points of solutions of some nonlinear problems

Khalil El Mehdi, Filomena Pacella (2002)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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In this Note we consider the following problem $\{\begin{matrix} - △ u = N (N - 2) u^{p_{ϵ}} - λ u & in Ω \\ u > 0 & in Ω \\ u = 0 & on \partial Ω . \end{matrix}$ where $Ω$ is a bounded smooth starshaped domain in $R^{N}$ , $N \geq 3$ , $p_{ϵ} = \frac{N + 2}{N - 2} - ϵ$ , $ϵ > 0$ , and $λ \geq 0$ . We prove that if $u_{ϵ}$ is a solution of Morse index $m > 0$ than $u_{ϵ}$ cannot have more than $m$ maximum points in $Ω$ for $ϵ$ sufficiently small. Moreover if $Ω$ is convex we prove that any solution of index one has only one critical point and the level sets are starshaped for $ϵ$ sufficiently small.

Some remarks on two-scale convergence and periodic unfolding

Jan Franců, Nils E M Svanstedt (2012)

Applications of Mathematics

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The paper discusses some aspects of the adjoint definition of two-scale convergence based on periodic unfolding. As is known this approach removes problems concerning choice of the appropriate space for admissible test functions. The paper proposes a modified unfolding which conserves integral of the unfolded function and hence simplifies the proofs and its application in homogenization theory. The article provides also a self-contained introduction to two-scale convergence and gives...