Accepted Paper
Inserted: 16 feb 2025
Last Updated: 16 feb 2025
Journal: Communications on Pure and Applied Analysis
Year: 2025
Abstract:
In this paper, we study the problem of finding the optimal hole among all subsets $A \subset \Omega$ that minimizes the $\alpha-$Holder seminorm $[u]_\alpha$ among all functions $u$ such that $u =1$ on $\partial\Omega$ and $u = 0$ on $A$, plus a penalization on the volume of $A$. For a given set $A \subset \Omega$, we will also characterize the function that minimizes $[u]_\alpha.$ In addition, we will study the limit when $p \to \infty$ of the “fractional” version of the Alt-Caffarelli problem.
Download: