\(\Gamma \)-convergence of some super quadratic functionals with singular weights
Fix an open bounded set \(\Omega\) of \(\mathbb R^3\) and fix \(\alpha= 2-p\) with \(p>2\). Let \(u\) be a scalar density function and let \(Tu\) denote its trace on \(\partial\Omega\). The distance function to the boundary is denoted by \(d(x,\partial\Omega)\). This paper studies the \(\Gamma\)-convergence of the following functional \(F_\varepsilon(u)\): \[ \varepsilon^{p-2} \int_\Omega |Du|^p \,d(x,\partial\Omega)^\alpha\,dx+ \varepsilon^{-\frac{p-2}{p-1}} \int_\Omega W(u)\,d(x,\partial\Omega)^{-\frac{\alpha}{p-1}}\,dx+ \varepsilon^{-1/2} \int_{\partial\Omega} V(Tu) \,d{\mathcal H}^2, \] where \({\mathcal H}^n\) is the \(n\)-dimensional Hausdorff measure, \(W\) and \(V\) are two nonnegative continuous functions vanishing at \(\alpha\), \(\beta\) and \(\alpha'\), \(\beta'\), respectively. It is shown that the singular limit of the energies \(F_\varepsilon\) leads to a coupled problem of bulk and surface phase transitions.
scientific article