Concentrated terms and varying domains in elliptic equations: Lipschitz case

TL;DR

This paper studies how solutions to nonlinear elliptic equations with oscillatory Lipschitz boundaries behave, showing they converge to a limit problem with transformed boundary conditions, capturing boundary oscillations.

Contribution

It establishes the convergence of solutions with oscillatory boundaries to a limit problem with flux boundary conditions, extending understanding of boundary oscillations in elliptic equations.

Findings

Solutions converge in H^1 to a limit problem

Boundary oscillations are captured by flux conditions

Upper semicontinuity of solutions family proved

Abstract

In this paper, we analyze the behavior of a family of solutions of a nonlinear elliptic equation with nonlinear boundary conditions, when the boundary of the domain presents a highly oscillatory behavior which is uniformly Lipschitz and nonlinear terms are concentrated in a region which neighbors the boundary domain. We prove that this family of solutions converges to the solutions of a limit problem in H^1 , an elliptic equation with nonlinear boundary conditions which captures the oscillatory behavior of the boundary and whose nonlinear terms are transformed into a flux condition on the boundary. Indeed, we show the upper semicontinuity of this family of solutions.