A singular perturbation approach to the Dirichlet-area minimisation problem

We study both one and two-phase minimisers of the Dirichlet-area energy $$E(v)={\int }_{B_1}\mathrm{\mid }\mathrm{\nabla }v{\mathrm{\mid }}^2+Per(\{v>0\},B_1)\mathrm{.}$$ In the two-phase case, we show that the energies $$E_{\epsilon }(v)={\int }_{B_1}\mathrm{\mid }\mathrm{\nabla }v{\mathrm{\mid }}^2+\frac{1}{\epsilon }W\left(\frac{v}{{\epsilon }^{1\mathrm{/}2}}\right),$$ $\mathrm{\Gamma }$-converge to $E$ as $\epsilon \to 0$, where $W$ is the double well potential extended by zero outside of $[-1,1]$ . As a consequence, we show that bounded local minimisers of $E_{\epsilon }$ converge to a local minimiser of $E$.