Elliptic and parabolic problems in thin domains with doubly weak oscillatory boundary

TL;DR

This paper studies the behavior of solutions to elliptic and parabolic problems in thin domains with oscillating boundaries, establishing spectral convergence and attractor properties as the domain shrinks.

Contribution

It introduces a novel analysis of elliptic and parabolic problems in thin domains with doubly oscillatory boundaries, including spectral convergence and attractor continuity.

Findings

Spectral convergence of elliptic operators as domains shrink

Upper semicontinuity of attractors and stationary states

Analysis of quasiperiodic boundary oscillations

Abstract

In this work we consider higher dimensional thin domains with the property that both boundaries, bottom and top, present oscillations of weak type. We consider the Laplace operator with Neumann boundary conditions and analyze the behavior of the solutions as the thin domains shrinks to a fixed domain $\omega\subset \R^n$. We obtain the convergence of the resolvent of the elliptic operators in the sense of compact convergence of operators, which in particular implies the convergence of the spectra. This convergence of the resolvent operators will allow us to conclude the global dynamics, in terms of the global attractors of a reaction diffusion equation in the thin domains. In particular, we show the upper semicontinuity of the attractors and stationary states. An important case treated is the case of a quasiperiodic situation, where the bottom and top oscillations are periodic but with period rationally independent.