$\Gamma$-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 ${\alpha }^{\prime },{\beta }^{\prime }$, 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.