Boundary regularity of a fourth order Alt-Caffarelli problem and applications to the minimization of the critical buckling load

TL;DR

This paper investigates the boundary regularity of a higher order Alt-Caffarelli problem related to shape optimization, providing new regularity results and an improved monotonicity formula for two-dimensional cases.

Contribution

It introduces new boundary regularity results for a fourth order problem and enhances the monotonicity formula with a novel epiperimetric inequality.

Findings

Full boundary regularity in two dimensions achieved

Boundary is analytic outside specific angles

Improved monotonicity formula with epiperimetric inequality

Abstract

We study a higher order analogue to the Alt-Caffarelli functional that arises in several shape optimization problems, among which the minimization of the critical buckling load of a clamped plate of fixed area. We obtain several regularity results up to the boundary in two dimensions, in particular we prove the full regularity of the boundary (analytic outside angles of opening $\approx 1.43\pi$) near any point of density less than 1 of the optimal shape. These results are based on the monotonicity formula discovered by Dipierro, Karakhanyan, and Valdinoci, which we improve with a new epiperimetric inequality.