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The Poisson problem in domains with Ahlfors regular boundary

Ariel Barton, Svitlana Mayboroda, Alberto Pacati

Abstract

We establish well posedness of the Poisson problem in weak local John domains, for linear second order elliptic equations with real coefficients, and with data in weighted Lebesgue spaces with a very broad range of acceptable parameters.

The Poisson problem in domains with Ahlfors regular boundary

Abstract

We establish well posedness of the Poisson problem in weak local John domains, for linear second order elliptic equations with real coefficients, and with data in weighted Lebesgue spaces with a very broad range of acceptable parameters.
Paper Structure (54 sections, 45 theorems, 534 equations, 6 figures)

This paper contains 54 sections, 45 theorems, 534 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The history of the Poisson problem and weighted averaged spaces
  3. Notation and function spaces
  4. Definitions
  5. A lemma concerning Ahlfors regular sets
  6. Whitney decompositions of open sets
  7. Basic properties of Whitney decompositions
  8. The basic lemma concerning the Whitney decomposition and weighted averaged Sobolev spaces
  9. The upper bound on the norm of F
  10. The lower bound on the norm of F for κ large enough
  11. Arbitrary κ
  12. Further results concerning the Whitney decomposition and weighted averaged Sobolev spaces
  13. Boundary values
  14. Definitions: weak local John domains and traces
  15. Bounding a function using a carrot path
  16. ...and 39 more sections

Key Result

Theorem 1.5

Let $\Omega\subseteq\mathbb{R}^{\mathfrak{n}}$, ${\mathfrak{n}}\geq 2$, be a connected open set. Suppose that $\partial\Omega$ is ${\mathfrak{d}}$-Ahlfors regular for some real number ${\mathfrak{d}}$ with $0<{\mathfrak{d}}<{\mathfrak{n}}$, and that $\mathbb{R}^{\mathfrak{n}}\setminus\Omega$ is unbo that satisfies for some constant $C$ depending on $p$, $s$, $\beta$, $A$, and $\Omega$, but not on

Figures (6)

  • Figure 1: Values of the parameters $(s,1/p)$ such that the estimate \ref{['eqn:Besov:estimate']} on solutions to the Poisson problem is valid. Because $\mathfrak{a}^*\geq 1-{\mathfrak{d}}$, the illustrated region is convex.
  • Figure 2: Values of the parameters $(s,1/p)$ such that the estimate \ref{['eqn:Besov:estimate']} is valid, given solvability of the $L^q$-Dirichlet problem for $L$ (on the left) or the $L^{q^*}$-Dirichlet problem for $L^*$ (on the right) with nontangential estimates. Because $\mathfrak{b}^*\geq 1-{\mathfrak{d}}+{\mathfrak{d}}/q$, the illustrated region is convex. In the common case that $q$ and $q^*$ both exist, the estimate \ref{['eqn:Besov:estimate']} is valid for values of $(s,1/p)$ as in the bottom picture.
  • Figure 3: If the condition \ref{['eqn:p<1:DGN']} is valid, then the conclusion \ref{['eqn:atom:Poisson:conclusion']} is valid whenever $(s,1/p)$ lies in the indicated triangle. The $s=1/2$, $p=2$ case of the bound \ref{['eqn:atom:Poisson:conclusion']} follows from Lemma \ref{['lem:beta']}.
  • Figure 4: If the condition \ref{['eqn:p<1:extrapolation']} is valid, then the conclusion \ref{['eqn:atom:Poisson:conclusion']} is valid whenever $(s,1/p)$ lies in the indicated region. All three of the dotted lines have slope $1/{\mathfrak{d}}$, and the region is nonempty (and $\mathfrak{a}^*\in (0,1)$) provided $\theta\in (0,1)$ and the point $(\theta,1/q)$ lies between the upper and lower dotted lines.
  • Figure 5: The regions mentioned in Section \ref{['sec:thm:Poisson']} in the case where ${\mathfrak{d}}\geq1-\alpha$ and ${\mathfrak{d}}\geq1-\alpha^*$
  • ...and 1 more figures

Theorems & Definitions (107)

  • Definition 1.2
  • Theorem 1.5
  • Remark 1.9
  • Remark 1.13
  • Theorem 1.17
  • Remark 1.19
  • Remark 2.3
  • Definition 2.4
  • Lemma 2.6
  • proof : Proof of Lemma \ref{['lem:close:to:Ahlfors']}
  • ...and 97 more