An anisotropic Alt-Caffarelli problem of higher order

TL;DR

This paper investigates a higher order anisotropic version of the Alt-Caffarelli problem in two dimensions, showing that smooth anisotropies do not alter the regularity of minimizers and extending fundamental estimates to anisotropic settings.

Contribution

It introduces a higher order anisotropic Alt-Caffarelli problem and proves that smooth anisotropies do not affect the optimal regularity of minimizers, extending classical estimates to anisotropic cases.

Findings

Smooth anisotropies do not affect $C^{2,1}$-regularity of minimizers.

Established an anisotropic version of Frehse's estimate for the bilaplacian.

Paves the way for studying higher order free boundary problems with anisotropy.

Abstract

We study a higher order version of the Alt-Caffarelli problem in two dimensions, where the Dirichlet energy is replaced by an anisotropic bending energy. This extends a previous study of the isotropic case in [41]. It turns out that smooth anisotropies do not affect the optimal $C^{2,1}$-regularity of minimizers. The proof requires an anisotropic version of an estimate by Frehse for the fundamental solution of the bilaplacian. This generalization paves the way for further studies of various free boundary problems of higher order.