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An anisotropic Alt-Caffarelli problem of higher order

Marius Müller

TL;DR

This work extends the Alt–Caffarelli free boundary framework to a higher-order, anisotropic setting in two dimensions by minimizing an elastic bending energy $\mathcal{E}(u)=\int_\Omega [\mathrm{div}(A\nabla u)]^2\,dx+|\{u>0\}|$ with smooth, uniformly elliptic $A$. A central contribution is the anisotropic Frehse observation, which yields a precise decomposition of $D^2u$ in terms of $Lu$, a kernel against the measure $\mu$ (the measure-valued part of $L^2u$), and a smooth remainder; this relies on an anisotropic analysis of Green's functions $G_{L}$ and $G_{L^2}$. Applying these ideas, the authors prove that smooth anisotropies do not affect the optimal $C^{2,1}$-regularity of minimizers, and they develop a comprehensive nodal-set analysis showing regularity of the free boundary and absence of singular points. They also derive a measure-valued equation for $\mu$ via inner variations, leading to an explicit density formula $\mu(A)=\int_{A\cap\{u=0\}} \frac{1}{2|\nabla u|}\,d\mathcal{H}^1$ and establish $u\in W^{3,p}_{\text{loc}}(\Omega)$ with $\{u=0\}$ being a $C^{2,\beta}$ surface. Overall, the paper lays a robust framework for higher-order anisotropic free boundary problems and their regularity theory.

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.

An anisotropic Alt-Caffarelli problem of higher order

TL;DR

This work extends the Alt–Caffarelli free boundary framework to a higher-order, anisotropic setting in two dimensions by minimizing an elastic bending energy with smooth, uniformly elliptic . A central contribution is the anisotropic Frehse observation, which yields a precise decomposition of in terms of , a kernel against the measure (the measure-valued part of ), and a smooth remainder; this relies on an anisotropic analysis of Green's functions and . Applying these ideas, the authors prove that smooth anisotropies do not affect the optimal -regularity of minimizers, and they develop a comprehensive nodal-set analysis showing regularity of the free boundary and absence of singular points. They also derive a measure-valued equation for via inner variations, leading to an explicit density formula and establish with being a surface. Overall, the paper lays a robust framework for higher-order anisotropic free boundary problems and their regularity theory.

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 -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.

Paper Structure

This paper contains 20 sections, 49 theorems, 258 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries about elliptic operators
  4. Supersolutions
  5. Green's function in 2D
  6. An anisotropic version of Frehse's observation
  7. Consequences for higher order measure-valued equations
  8. Variational methods
  9. Existence of minimizers
  10. Estimates for minimizers
  11. Regularity away from the free boundary
  12. Variational inequalities
  13. Semiconvexity
  14. Analysis of the nodal set
  15. Good points
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $u \in W^{2,2}_{loc}(\Omega)$ be such that for some finite Radon measure $\mu$ one has $L^2u = \mu$ in the sense of distributions, i.e. Then, there exists a Borel measurable function $K: \Omega \times \Omega \rightarrow \mathbb{R}^{2\times 2}_{sym}$ which is locally bounded and smooth in $(\Omega \times \Omega) \setminus \{(x,x): x \in \Omega \}$ and $H \in C^\infty(\Omega; \mathbb{R}^{2\time

Theorems & Definitions (101)

  • Theorem 1.1: Anisotropic Frehse Observation
  • Theorem 1.2: Higher Order Alt-Caffarelli Problem
  • Definition 2.1
  • Lemma 2.2: Regularity of weak supersolutions, see AQ02Veron
  • proof : Sketch of Proof.
  • Lemma 2.3: BH15
  • Lemma 2.4
  • proof
  • Corollary 2.5: Strong maximum principle
  • proof
  • ...and 91 more