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Boundary regularity for the polyharmonic Dirichlet problem

Antoine Lemenant, Rémy Mougenot

Abstract

In this paper we prove that any solution of the $m$-polyharmonic Poisson equation in a Reifenberg-flat domain with homogeneous Dirichlet boundary condition, is $\mathscr{C}^{m-1,α}$ regular up to the boundary. To achieve this result we extend the Nirenberg method of translations to operators of arbitrary order, and then use some Mosco-convergence tools developped in a previous paper.

Boundary regularity for the polyharmonic Dirichlet problem

Abstract

In this paper we prove that any solution of the -polyharmonic Poisson equation in a Reifenberg-flat domain with homogeneous Dirichlet boundary condition, is regular up to the boundary. To achieve this result we extend the Nirenberg method of translations to operators of arbitrary order, and then use some Mosco-convergence tools developped in a previous paper.

Paper Structure

This paper contains 15 sections, 17 theorems, 198 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Multi-index and Leibniz formula
  4. Elliptic operator of order $2m$
  5. Weak Formulation
  6. Matrix formulation
  7. Poly-harmonic operators
  8. Decomposition of Elliptic operators
  9. Poincaré type inequalities
  10. Finite difference tools
  11. Campanato spaces
  12. Interior regularity by the translation method
  13. Boundary regularity in the case of a hyperplane
  14. Energy decay for a Reifenberg-flat boundary
  15. Boundary regularity in a Reifenberg flat domain

Key Result

Theorem 1.1

Let $\alpha \in (0,1)$ and $q\geq 2$ be such that $mq\geq N$ if $2m<N$, and $q=2$ otherwise. There exists $\varepsilon_0 \in (0,1)$ and $r_0 \in (0,1]$ such that for every $(\varepsilon_0,r_0)-$Reifenberg-flat domain $\Omega\subset \mathbb{R}^N$, for every function $f\in L^q(\Omega)$, if $u \in H^m_ where $C>0$ depends on $N$, $A$, $\alpha$, $\Omega$ and $m$.

Theorems & Definitions (32)

  • Definition 1.1: Reifenberg flat domain
  • Theorem 1.1
  • Theorem 1.2: Regularity with flat boundary
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Proposition 2.1
  • proof
  • ...and 22 more