$\Gamma$-convergence of some super quadratic functionals with singular weights

Abstract

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