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Variational analysis of discrete Dirichlet problems in periodically perforated domains

Giuliana Fusco

Abstract

In this paper we study the asymptotic behavior of a family of discrete functionals as the lattice size, $\varepsilon>0$, tends to zero. We consider pairwise interaction energies satisfying $p$-growth conditions, $p<d$, $d$ being the dimension of the reference configuration, defined on discrete functions subject to Dirichlet conditions on a $δ$-periodic array of small squares of side $r_δ\sim δ^{d/d-p}$. Our analysis is performed in the framework of $Γ$-convergence and we prove that, in the regime $\varepsilon=o(r_δ)$, the discrete energy and their continuum counterpart share the same $Γ$-limit and the effect of the constraints leads to a capacitary term in the limit energy as in the classical theory of periodically perforated domains for local integral functionals.

Variational analysis of discrete Dirichlet problems in periodically perforated domains

Abstract

In this paper we study the asymptotic behavior of a family of discrete functionals as the lattice size, , tends to zero. We consider pairwise interaction energies satisfying -growth conditions, , being the dimension of the reference configuration, defined on discrete functions subject to Dirichlet conditions on a -periodic array of small squares of side . Our analysis is performed in the framework of -convergence and we prove that, in the regime , the discrete energy and their continuum counterpart share the same -limit and the effect of the constraints leads to a capacitary term in the limit energy as in the classical theory of periodically perforated domains for local integral functionals.

Paper Structure

This paper contains 10 sections, 14 theorems, 167 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. Setting of the problem and the main result
  4. Preliminary results
  5. Supporting results
  6. Gagliardo-Nirenberg-Sobolev type inequality
  7. Two technical lemmas
  8. Approximating capacitary energy densities
  9. Asymptotics of the approximating capacitary energy density
  10. Proof of Theorem \ref{['MAIN']}

Key Result

Theorem 3.3

Let $\mathcal{F}_{\varepsilon}$ be defined by (unconstrained functional), with $f$ satisfying assumptions (H) and (G). Then $(\mathcal{F}_{\varepsilon})$$\Gamma$-converges with respect to the $L^{p}(\Omega;\mathbb{R}^{m})$-topology to the functional $\mathcal{F}:L^{p}(\Omega;\mathbb{R}^{m})\rightarr where $f_{\hom}:\mathbb{R}^{m\times d}\rightarrow [0,+\infty)$ is given by the following homogeniza

Figures (1)

  • Figure 1:

Theorems & Definitions (27)

  • Remark 3.1
  • Remark 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Remark 3.5
  • Lemma 4.1: alicic, Lemma 3.6
  • Remark 4.2
  • Remark 4.3
  • Lemma 4.4: AS, Lemma 2
  • Lemma 4.5: AS, Lemma 3
  • ...and 17 more