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Homogenization of supremal functionals in the vectorial case (via $L^p$-approximation)

Lorenza D'Elia, Michela Eleuteri, Elvira Zappale

Abstract

We propose a homogenized supremal functional rigorously derived via $L^p$-approximation by functionals of the type $\underset{x\inΩ}{\mbox{ess-sup}}\hspace{0.03cm} f\left(\frac{x}{\varepsilon}, Du\right)$, when $Ω$ is a bounded open set of $\mathbb R^n$ and $u\in W^{1,\infty}(Ω;\mathbb R^d)$. The homogenized functional is also deduced directly in the case where the sublevel sets of $f(x,\cdot)$ satisfy suitable convexity properties, as a corollary of homogenization results dealing with pointwise gradient constrained integral functionals.

Homogenization of supremal functionals in the vectorial case (via $L^p$-approximation)

Abstract

We propose a homogenized supremal functional rigorously derived via -approximation by functionals of the type , when is a bounded open set of and . The homogenized functional is also deduced directly in the case where the sublevel sets of satisfy suitable convexity properties, as a corollary of homogenization results dealing with pointwise gradient constrained integral functionals.
Paper Structure (14 sections, 22 theorems, 189 equations, 1 figure)

This paper contains 14 sections, 22 theorems, 189 equations, 1 figure.

Table of Contents

  1. Introduction and statement of the problem
  2. Notation and preliminary results
  3. General notations and basic facts
  4. $\Gamma$-convergence
  5. The $\boldsymbol{L^p}$-approximation
  6. The cell formula
  7. Homogenization of unbounded functionals
  8. Measure representation results of $\Gamma$-limits
  9. Finiteness conditions
  10. Representation on Linear Functions and a blow-up condition
  11. Representation on Sobolev spaces
  12. The proof of Theorem \ref{['Proposition 12.6.1']}
  13. Homogenization
  14. Appendix

Key Result

Proposition 2.1

Let $X$ be a metric space and let $\varphi_{k}: X \to \mathbb R \cup \{\pm \infty\}$, for every $k\in \mathbb{N}$. Then $\{\varphi_k\}_{k}$$\Gamma$-converges to $\varphi$ with respect to the strong topology of $X$ (and we write $\Gamma(X)\hbox{-}\lim_{k\to +\infty}\varphi_{k}=\varphi$) if and only i

Figures (1)

  • Figure :

Theorems & Definitions (46)

  • Proposition 2.1: DM93
  • Definition 2.2
  • Proposition 2.3
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Lemma 3.4
  • proof
  • Proposition 3.5
  • ...and 36 more