A singular perturbation approach to the Dirichlet-area minimisation problem

TL;DR

This paper investigates the minimization of a combined Dirichlet and perimeter energy, establishing a singular perturbation approach that demonstrates convergence of approximate energies to the original problem.

Contribution

It introduces a singular perturbation method for the Dirichlet-area minimization problem and proves the convergence of approximate energies to the original energy as the perturbation parameter tends to zero.

Findings

$ ext{E}_ ext{ε}$ $ ext{Γ}$-converges to $E$ as $ ext{ε} o 0$

Bounded local minimisers of $ ext{E}_ ext{ε}$ converge to local minimisers of $E$

The approach links two-phase minimisers with singular perturbation analysis.

Abstract

We study both one and two-phase minimisers of the Dirichlet-area energy $$E(v) = \int_{B_1} \vert\nabla v\vert^2 + Per(\{v>0\},B_1).$$ In the two-phase case, we show that the energies $$E_{\varepsilon}(v) = \int_{B_1}\vert\nabla v\vert^2 + \frac{1}{\varepsilon}W\left(\frac{v}{\varepsilon^{1/2}}\right),$$ $\Gamma$-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 $E$.