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In this paper, we use $Γ$ -convergence techniques to study the following variational problem $S_{ε}^{F} (Ω) : = sup \{ε^{- 2^{*}} \int_{Ω} F (u) d x : \int_{Ω} {| \nabla u |}^{2} d x \leq ε^{2}, u = 0 on \partial Ω\},$ where $0 \leq F (t) \leq {| t |}^{2^{*}}$ , with $2^{*} = \frac{2 n}{n - 2}$ , and $Ω$ is a bounded domain of $ℝ^{n}$ , $n \geq 3$ . We obtain a $Γ$ -convergence result, on which one can easily read the usual concentration phenomena arising in critical growth problems. We extend the result to a non-homogeneous version of problem $S_{ε}^{F} (Ω)$ . Finally, a second order expansion in $Γ$ -convergence permits to identify the concentration points of the maximizing sequences, also in some non-homogeneous case.

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

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 p-growth assumptions on the bulk energy, and the assumption that the surface energy satisfies p-growth upper bound,...

Γ-convergence of functionals on divergence-free fields

Nadia Ansini, Adriana Garroni (2007)

ESAIM: Control, Optimisation and Calculus of Variations

We study the stability of a sequence of integral functionals on divergence-free matrix valued fields following the direct methods of Γ-convergence. We prove that the Γ-limit is an integral functional on divergence-free matrix valued fields. Moreover, we show that the Γ-limit is also stable under volume constraint and various type of boundary conditions.

$ξ$ -устойчивость классов отображений и системы линейных уравнений с частными производными.

Н.С. Дaирбеков, А.П. Копылов (1985)

Sibirskij matematiceskij zurnal

$Σ$ -convergence.

Nguetseng, Gabriel, Svanstedt, Nils (2011)

Banach Journal of Mathematical Analysis [electronic only]

$Σ$ -convergence of nonlinear monotone operators in perforated domains with holes of small size

Jean Louis Woukeng (2009)

Applications of Mathematics

This paper is devoted to the homogenization beyond the periodic setting, of nonlinear monotone operators in a domain in $ℝ^{N}$ with isolated holes of size $ε^{2}$ ( $ε > 0$ a small parameter). The order of the size of the holes is twice that of the oscillations of the coefficients of the operator, so that the problem under consideration is a reiterated homogenization problem in perforated domains. The usual periodic perforation of the domain and the classical periodicity hypothesis on the coefficients of the operator...

$φ$ -Laplacian BVPs with linear bounded operator conditions

Kamal Bachouche, Smaïl Djebali, Toufik Moussaoui (2012)

Archivum Mathematicum

The aim of this paper is to present new existence results for $φ$ -Laplacian boundary value problems with linear bounded operator conditions. Existence theorems are obtained using the Schauder and the Krasnosel’skii fixed point theorems. Some examples illustrate the results obtained and applications to multi-point boundary value problems are provided.

Ψ-pseudodifferential operators and estimates for maximal oscillatory integrals

Carlos E. Kenig, Wolfgang Staubach (2007)

Studia Mathematica

We define a class of pseudodifferential operators with symbols a(x,ξ) without any regularity assumptions in the x variable and explore their $L^{p}$ boundedness properties. The results are applied to obtain estimates for certain maximal operators associated with oscillatory singular integrals.

ω-ultradistributions and their application to operator theory

Ioana Ciorǎnescu, László Zsidó (1982)

Banach Center Publications

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