A singular perturbation approach to the Dirichlet-area minimisation problem

Abstract

We study both one and two-phase minimisers of the Dirichlet-area energy

E (v) = \int_{B_{1}} ∣\nabla v ∣^{2} + Per ({v > 0}, B_{1}) .

In the two-phase case, we show that the energies

E_{ε} (v) = \int_{B_{1}} ∣\nabla v ∣^{2} + \frac{1}{ε} W (\frac{v}{ε ^{1/2}}),

$Γ$ -converge to $E$ as $ε \to 0$ , where $W$ is the double well potential extended by zero outside of $[- 1, 1]$ . As a consequence, we show that bounded local minimisers of $E_{ε}$ converge to a local minimiser of $E$ .

A singular perturbation approach to the Dirichlet-area minimisation problem

Anthony Salib

Georg S. Weiss

Abstract