Global minimizers for a two-sided biharmonic Alt-Caffarelli problem

TL;DR

This paper investigates the global minimizers of a biharmonic version of the Alt-Caffarelli problem, revealing which solutions are minimal across dimensions and highlighting differences from the one-sided case.

Contribution

It identifies classes of global minimizers for the two-sided biharmonic Alt-Caffarelli problem and shows they differ significantly from the one-sided case, including the absence of PDE satisfaction.

Findings

Half-space solutions are global minimizers in the two-sided case.

Three categories of minimizers persist in all dimensions.

Minimizers generally do not satisfy a PDE, unlike in the one-sided case.

Abstract

We study global minimizers of biharmonic analogues of the Alt-Caffarelli functional. It turns out that half-space solutions are global minimizers for the two-sided Alt-Caffarelli functional, but not in the one-sided case. In addition, we identify a further class of global minimizers, all of which have constant Laplacian. Recent work by J. Lamboley and M. Nahon reduces potential global minimizers in dimension two to four possible categories. Our work shows that three of these categories persist in any dimension and are in fact global minimizers. Moreover, we show that minimizers of the two-sided biharmonic Alt-Caffarelli problem do in general not satisfy a partial differential equation, not even with a signed measure as right-hand-side. This is in sharp contrast to the corresponding one-sided problem.