Submitted Paper

Inserted: 13 nov 2021
Last Updated: 13 nov 2021

Year: 2021

Abstract:

We study the regularity of the interface for optimal energy configurations of functionals involving bulk energies with an additional perimeter penalization of the interface. It is allowed the dependence on $(x,u)$ for the bulk energy. For a minimal configuration $(E,u)$, the H\"{o}lder continuity of $u$ is well known. We give an estimate for the singular set of the boundary $\partial E$. Namely we show that the Hausdorff dimension of the singular set is strictly smaller than $n-1$.

Keywords: free boundary, minimal surfaces, perimeter penalization, volume constraint

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