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$L^p$-solvability of boundary value problems for the Laplacian in locally flat unbounded domains

Ignasi Guillén-Mola

TL;DR

This work proves $L^p$ solvability for the Dirichlet problem and the $L^{p'}$ solvability for the Neumann problem for the Laplacian in $2$-sided chord-arc domains with unbounded boundary, under large-scale flatness and a finite number of 'bad' boundary balls. The authors implement a layer-potential framework by showing the boundary double layer operator $K$ is a compact perturbation of a small-norm operator, then invoke Fredholm theory to obtain invertibility of $\tfrac{1}{2}{\rm Id}+K$ and its adjoint, establishing solvability and uniqueness. A central innovation is the decomposition of $K$ into small, intermediate, and large-scale components, with compactness arising from intermediate scales and small/large-scale parts controlled via Semmes’ decomposition and a localized good-lambda strategy. The large-scale analysis uses a Lipschitz-graph approximation and a meticulous handling of the unit normal vector’s $\mathrm{BMO}$-type oscillation and Jones’ $\beta$-numbers to secure a bound that feeds into the Fredholm framework. Collectively, the results extend classical layer-potential solvability from bounded to certain unbounded geometries and pave the way for extrapolation to more general divergence-form operators.

Abstract

We establish the solvability of the $L^p$-Dirichlet and $L^{p^\prime}$-Neumann problems for the Laplacian for $p\in (\frac{n}{n-1}-\varepsilon,\frac{2n}{n-1}]$ for some $\varepsilon>0$ in $2$-sided chord-arc domains with unbounded boundary that is sufficiently flat at large scales and outward unit normal vector whose oscillation fails to be small only at finitely many dyadic boundary balls.

$L^p$-solvability of boundary value problems for the Laplacian in locally flat unbounded domains

TL;DR

This work proves solvability for the Dirichlet problem and the solvability for the Neumann problem for the Laplacian in -sided chord-arc domains with unbounded boundary, under large-scale flatness and a finite number of 'bad' boundary balls. The authors implement a layer-potential framework by showing the boundary double layer operator is a compact perturbation of a small-norm operator, then invoke Fredholm theory to obtain invertibility of and its adjoint, establishing solvability and uniqueness. A central innovation is the decomposition of into small, intermediate, and large-scale components, with compactness arising from intermediate scales and small/large-scale parts controlled via Semmes’ decomposition and a localized good-lambda strategy. The large-scale analysis uses a Lipschitz-graph approximation and a meticulous handling of the unit normal vector’s -type oscillation and Jones’ -numbers to secure a bound that feeds into the Fredholm framework. Collectively, the results extend classical layer-potential solvability from bounded to certain unbounded geometries and pave the way for extrapolation to more general divergence-form operators.

Abstract

We establish the solvability of the -Dirichlet and -Neumann problems for the Laplacian for for some in -sided chord-arc domains with unbounded boundary that is sufficiently flat at large scales and outward unit normal vector whose oscillation fails to be small only at finitely many dyadic boundary balls.

Paper Structure

This paper contains 24 sections, 47 theorems, 280 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries and definitions
  3. Notation
  4. ADR, UR, Uniform, NTA and CAD
  5. The nontangential maximal operator and boundary dyadic cubes in ADR domains
  6. Compact operators
  7. Calderón-Zygmund operators and the Riesz transform
  8. The double layer potential
  9. Proof of \ref{['thm:double layer potential is almost compact']} and consequences
  10. The compact operator: Proof of \ref{['trucated and intermediate scales is compact']}
  11. The double layer potential in small scales. Proof of \ref{['lemma:double layer potentials in small scales has small norm', 'lemma:double layer potentials far from bad ball has small norm']}
  12. The double layer potential in large scales. Proof of \ref{['large scale bounded by bmo norm']}
  13. Relation between unit normal vectors
  14. Notation and parameters for the proof
  15. Reduction to a good lambda inequality
  16. ...and 9 more sections

Key Result

Theorem 1.2

Let $\Omega\subset {\mathbb R}^{n+1}$ be a $\delta$-$(s,S;R)$ domain. There is $\varepsilon_D=\varepsilon_D (n,\text{CAD})\in (0,\frac{1}{n-1})$ such that for every $p_0\in (\frac{n}{n-1}-\varepsilon_D,\frac{2n}{n-1}]$ there exists $\delta_0=\delta_0(n,p_0,\text{CAD})>0$ such that if $\delta\leq \de is the solution of $(D_{p_0})$. Furthermore, there exists $\varepsilon>0$ such that $(D_p)$ is solv

Theorems & Definitions (100)

  • Definition 1.1: $\delta$-$(s,S;R)$ domain
  • Theorem 1.2: $L^p$-Dirichlet problem
  • Theorem 1.3: $L^p$-Neumann problem
  • Theorem 1.4
  • Definition 2.1: UR set
  • Definition 2.2: Corkscrew ball conditions
  • Remark 2.3: The geometric measure theoretic outward unit normal vector $\nu$
  • Definition 2.4: Harnack chain condition
  • Definition 2.5: Uniform domain
  • Definition 2.6: NTA domain
  • ...and 90 more