On some Variational Problems set on domains tending to infinity

Abstract

Let ${\Omega }_{\ell }=\ell {\omega }_1\times {\omega }_2$ where ${\omega }_1\subset {\mathbb{R}}^p$ and ${\omega }_2\subset {\mathbb{R}}^{n-p}$ are assumed to be open and bounded. We consider the following minimization problem: $$E_{{\Omega }_{\ell }}(u_{\ell })=\underset{u\in W_0^{1,q}({\Omega }_{\ell })}{\min }{E}_{{\Omega }_{\ell }}(u)$$ where $E_{Ω_\ell}(u) = \int_{Ω_\ell}F(\grad u)-fu$, $F$ is a convex function and $f\in L^{q^{\prime }}({\omega }_2)$. We are interested in studying the asymptotic behavior of the solution $u_{\ell }$ as $\ell$ tends to infinity.

to appear in DCDS-Series A