On some Variational Problems set on domains tending to infinity

TL;DR

This paper investigates the asymptotic behavior of solutions to variational problems defined on domains expanding infinitely in certain directions, with the goal of understanding how solutions evolve as the domain size increases.

Contribution

It provides an analysis of the limiting behavior of minimizers for variational problems on unbounded domains formed by expanding bounded regions.

Findings

Characterization of the limit of solutions as domain size tends to infinity

Identification of conditions for convergence of minimizers

Insights into the influence of domain geometry on variational solutions

Abstract

Let $\Omega_\ell = \ell\omega_1 \times \omega_2$ where $\omega_1 \subset \R^p$ and $\omega_2 \subset \R^{n-p}$ are assumed to be open and bounded. We consider the following minimization problem: $$E_{\Omega_\ell}(u_\ell) = \min_{u\in W_0^{1,q}(\Omega_\ell)}E_{\Omega_\ell}(u)$$ where $E_{\Omega_\ell}(u) = \int_{\Omega_\ell}F(\grad u)-fu$, $F$ is a convex function and $f\in L^{q'}(\omega_2)$. We are interested in studying the asymptotic behavior of the solution $u_\ell$ as $\ell$ tends to infinity.