Viscosity method for Homogenization of Parabolic Nonlinear Equations in Perforated Domains

TL;DR

This paper introduces a viscosity method for homogenizing nonlinear parabolic equations in perforated domains with oscillating obstacles, revealing how obstacle capacity influences the limit behavior of solutions.

Contribution

It develops a novel viscosity-based homogenization approach for nonlinear parabolic equations constrained by oscillating obstacles in perforated domains.

Findings

Critical decay rates for obstacle capacity determine nontrivial limits.

Limit solutions satisfy a homogenized equation with obstacle effects in viscosity sense.

Homogenization results depend on obstacle geometry and distribution.

Abstract

In this paper, we develop a viscosity method for Homogenization of Nonlinear Parabolic Equations constrained by highly oscillating obstacles or Dirichlet data in perforated domains. The Dirichlet data on the perforated domain can be considered as a constraint or an obstacle. Homogenization of nonlinear eigen value problems has been also considered to control the degeneracy of the Porous medium equation in perforated domains. For the simplicity, we consider obstacles that consist of cylindrical columns distributed periodically and perforated domains with punctured balls. If the decay rate of the capac- ity of columns or the capacity of punctured ball is too high or too small, the limit of u\k{o} will converge to trivial solutions. The critical decay rates of having nontrivial solution are obtained with the construction of barriers. We also show the limit of u\k{o} satisfies a homogenized equation with a term showing the effect of the highly oscillating obstacles or perforated domain in viscosity sense.