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Homogenization of an obstacle problem with highly oscillating coefficients and obstacles

Sunghoon Kim, Ki-Ahm Lee, Se-Chan Lee, Minha Yoo

Abstract

We develop the viscosity method for the homogenization of an obstacle problem with highly oscillating obstacles. The associated operator, in non-divergence form, is linear and elliptic with variable coefficients. We first construct a highly oscillating corrector, which captures the singular behavior of solutions near periodically distributed holes of critical size. We then prove the uniqueness of a critical value that encodes the coupled effects of oscillations in both the coefficients and the obstacles.

Homogenization of an obstacle problem with highly oscillating coefficients and obstacles

Abstract

We develop the viscosity method for the homogenization of an obstacle problem with highly oscillating obstacles. The associated operator, in non-divergence form, is linear and elliptic with variable coefficients. We first construct a highly oscillating corrector, which captures the singular behavior of solutions near periodically distributed holes of critical size. We then prove the uniqueness of a critical value that encodes the coupled effects of oscillations in both the coefficients and the obstacles.

Paper Structure

This paper contains 8 sections, 20 theorems, 176 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Almost uniform convergence
  4. Correctors and critical values
  5. Existence of periodic correctors
  6. Uniqueness of the critical value b0
  7. Proof of Theorem 1.1
  8. Homogenization for non-critical hole sizes

Key Result

Theorem 1.1

Let $u_{\varepsilon}$ be the least viscosity supersolution of problem-Le.

Theorems & Definitions (36)

  • Theorem 1.1
  • Remark 2.1: Bau84
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4: HK20
  • Lemma 2.5: DKL23
  • Theorem 2.6
  • proof
  • Lemma 3.1: Discrete gradient estimate
  • proof
  • ...and 26 more