$Γ$-convergence of some super quadratic functionals with singular weights

Abstract

We study the $Γ$-convergence of the following functional ($p>2$) $$ F_ε(u):=ε^{p-2}\int_Ω|Du|^p d(x,\partial Ω)^{a}dx+\frac{1}{ε^{\frac{p-2}{p-1}}}\int_ΩW(u) d(x,\partial Ω)^{-\frac{a}{p-1}}dx+\frac{1}{\sqrtε}\int_{\partialΩ}V(Tu)d\mathcal{H}^2, $$ where $Ω$ is an open bounded set of $\mathbb{R}^3$ and $W$ and $V$ are two non-negative continuous functions vanishing at $α, β$ and $α', β'$, respectively. In the previous functional, we fix $a=2-p$ and $u$ is a scalar density function, $Tu$ denotes its trace on $\partialΩ$, $d(x,\partial Ω)$ stands for the distance function to the boundary $\partial\Om$. We show that the singular limit of the energies $F_ε$ leads to a coupled problem of bulk and surface phase transitions.