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

Hans-Christoph Grunau, Marius Müller

TL;DR

This paper analyzes global minimizers of the two-sided biharmonic Alt-Caffarelli functional $\mathcal{F}$ in $\mathbb{R}^n$ and identifies two robust global minimizer families: Type I half-space minimizers and Type II minimizers with constant $\Delta u$, the latter global when $|\Delta u|\ge1$. It proves Type I minimizers are globally minimal via a higher-order Velichkov-type inequality and an extension argument, while Type II minimizers yield infinitely many two-homogeneous global minimizers; the results also confirm that three categories can persist in any dimension, with some potential angular minimizers known in low dimensions. A key finding is that two-sided minimizers do not satisfy a PDE with a measure on the right-hand side; this is demonstrated through explicit computations and regularity considerations, highlighting a sharp qualitative difference from the one-sided problem. The work clarifies blow-up behavior, regularity, and the structure of global minimizers for the biharmonic Alt-Caffarelli problem and outlines limits of PDE-based descriptions for these minimizers.

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.

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

TL;DR

This paper analyzes global minimizers of the two-sided biharmonic Alt-Caffarelli functional in and identifies two robust global minimizer families: Type I half-space minimizers and Type II minimizers with constant , the latter global when . It proves Type I minimizers are globally minimal via a higher-order Velichkov-type inequality and an extension argument, while Type II minimizers yield infinitely many two-homogeneous global minimizers; the results also confirm that three categories can persist in any dimension, with some potential angular minimizers known in low dimensions. A key finding is that two-sided minimizers do not satisfy a PDE with a measure on the right-hand side; this is demonstrated through explicit computations and regularity considerations, highlighting a sharp qualitative difference from the one-sided problem. The work clarifies blow-up behavior, regularity, and the structure of global minimizers for the biharmonic Alt-Caffarelli problem and outlines limits of PDE-based descriptions for these minimizers.

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.
Paper Structure (13 sections, 18 theorems, 73 equations)

This paper contains 13 sections, 18 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Application: Blow-Up-Analysis
  3. Structure of the article
  4. Preliminaries
  5. Properties of global minimizers
  6. Type I: Half-space minimizers
  7. Proof of Minimality
  8. Nonminimality for the one-sided functional
  9. Type II: Minimizers with constant Laplacian
  10. Nonextendability of categories (I) and (II)
  11. No measure-valued equation for minimizers
  12. Distributional equations for explicit solutions
  13. Regularity of measure-valued solutions

Key Result

Lemma 1

Let $\Omega \subset \mathbb{R}^n$ be open and $u_0 \in C^\infty(\Omega)$. Further let $u \in H^2(\Omega)$ be a minimizer of $\mathcal{F}(\cdot, \Omega)$ in $\mathscr{D}(u_0):= \{ v \in H^2(\Omega) : v -u_0 \in H_0^2(\Omega) \}.$ Then $u \in C^{1,\alpha}(\Omega)$ and $u$ is smooth and biharmonic on $

Theorems & Definitions (40)

  • Definition 1
  • Definition 2: Velichkov
  • Definition 3
  • Lemma 1: GMMathAnn
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 30 more