A singular perturbation approach to the Dirichlet-area minimisation problem

We study both one and two-phase minimisers of the Dirichlet-area energy $$ E(v) = \int\_{B\_1} |\nabla v|^2 + \mathrm{Per}({v>0},B\_1). $$ In the two-phase case, we show that the energies $$ E\_{\varepsilon}(v) = \int\_{B\_1}|\nabla v|^2 + \frac{1}{\varepsilon}W\Big(\frac{v}{\varepsilon^{1/2}}\Big), $$ $\Gamma$-converge to $E$ as $\varepsilon \to 0$, where $W$ is the double well potential extended by zero outside of $\[-1,1]$. As a consequence, we show that bounded local minimisers of $E\_{\varepsilon}$ converge to a local minimiser of $E$.