Hölder continuity in a shape-optimization problem with perimeter

Abstract

Here, we prove Hölder continuity of the state function in a shape-optimization problem arising, for instance, in the electromagnetic shaping of a liquid metal. The equilibrium state is obtained as minimizing the total energy, which is given in the form $$ \mathcal{E}_\lambda(\Omega) = J(\Omega) + P(\Omega) + \lambda ||\Omega| - m| , $$ where $\Omega$ is the domain occupied by the liquid, $P(\Omega)$ and $|\Omega|$ being respectively the perimeter and the Lebesgue measure of $\Omega$. We prove that the state function associated with the optimal shape is $\frac{1}{2}$-H\"older continuous, the main difficulty coming from the fact that the state function is not assumed to be non-negative.