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Well-posedness of Dirichlet boundary value problems for reflected fractional $p$-Laplace-type inhomogeneous equations in compact doubling metric measure spaces

Josh Kline, Feng Li, Nageswari Shanmugalingam

TL;DR

The work develops a rigorous framework for the well-posedness of a reflected fractional $p$-Laplace-type Dirichlet problem on a compact doubling metric measure space $Z$, viewed as the boundary of a higher-dimensional uniform domain. Central to the approach is a Dirichlet-to-Neumann type operator $\mathcal{E}_T$ defined via $p$-harmonic extensions on the uniform domain, together with a robust trace theory relating Besov spaces on $\partial Z$ to Newton-Sobolev spaces on the filling. The authors establish existence, uniqueness, a comparison principle, and stability with respect to both boundary data and inhomogeneity data; they also prove interior Hölder regularity and a Kellogg-type boundary regularity property for the homogeneous problem. The results rely on trace/extension operators between Besov and Newton-Sobolev spaces, a variational energy framework, and hyperbolic fillings, enabling analysis in nonsmooth settings including fractal-like spaces. This advances nonlocal Dirichlet problems in spaces where Poincaré inequalities may fail and extends Dirichlet-to-Neumann operator theory to highly irregular boundaries with codimensional measures.

Abstract

In this paper we consider the setting of a locally compact, non-complete metric measure space $(Z,d,ν)$ equipped with a doubling measure $ν$, under the condition that the boundary $\partial Z:=\overline{Z}\setminus Z$ (obtained by considering the completion of $Z$) supports a Radon measure $π$ which is in a $σ$-codimensional relationship to $ν$ for some $σ>0$. We explore existence, uniqueness, comparison property, and stability properties of solutions to inhomogeneous Dirichlet problems associated with certain nonlinear nonlocal operators on $Z$. We also establish interior regularity of solutions when the inhomogeneity data is in an $L^q$-class for sufficiently large $q>1$, and verify a Kellogg-type property when the inhomogeneity data vanishes and the Dirichlet data is continuous.

Well-posedness of Dirichlet boundary value problems for reflected fractional $p$-Laplace-type inhomogeneous equations in compact doubling metric measure spaces

TL;DR

The work develops a rigorous framework for the well-posedness of a reflected fractional -Laplace-type Dirichlet problem on a compact doubling metric measure space , viewed as the boundary of a higher-dimensional uniform domain. Central to the approach is a Dirichlet-to-Neumann type operator defined via -harmonic extensions on the uniform domain, together with a robust trace theory relating Besov spaces on to Newton-Sobolev spaces on the filling. The authors establish existence, uniqueness, a comparison principle, and stability with respect to both boundary data and inhomogeneity data; they also prove interior Hölder regularity and a Kellogg-type boundary regularity property for the homogeneous problem. The results rely on trace/extension operators between Besov and Newton-Sobolev spaces, a variational energy framework, and hyperbolic fillings, enabling analysis in nonsmooth settings including fractal-like spaces. This advances nonlocal Dirichlet problems in spaces where Poincaré inequalities may fail and extends Dirichlet-to-Neumann operator theory to highly irregular boundaries with codimensional measures.

Abstract

In this paper we consider the setting of a locally compact, non-complete metric measure space equipped with a doubling measure , under the condition that the boundary (obtained by considering the completion of ) supports a Radon measure which is in a -codimensional relationship to for some . We explore existence, uniqueness, comparison property, and stability properties of solutions to inhomogeneous Dirichlet problems associated with certain nonlinear nonlocal operators on . We also establish interior regularity of solutions when the inhomogeneity data is in an -class for sufficiently large , and verify a Kellogg-type property when the inhomogeneity data vanishes and the Dirichlet data is continuous.
Paper Structure (19 sections, 21 theorems, 140 equations)

This paper contains 19 sections, 21 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Metric and measure-theoretic notions
  4. Potential theory
  5. Uniform domains
  6. Compact doubling metric measure spaces as boundaries of uniform domains of globally controlled geometry
  7. Cheeger differential structure and the definition of differential $\nabla f$
  8. Standing assumptions and statement of the Dirichlet problem for fractional operators on $Z$
  9. Examples
  10. Besov functions with zero trace
  11. Traces of Besov spaces on $Z$, at $\partial Z$.
  12. Existence of solutions to the inhomogeneous Dirichlet problem
  13. Stability
  14. Stability with respect to the Dirichlet boundary data
  15. Stability with respect to the inhomogeneity data
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $0<\theta<1$ and $1<p<\infty$, and let $u\in B^\theta_{p,p}(\overline Z,\nu)$. Furthermore, if $p>\max\{1,\sigma/\theta\}$ and $\pi$ is $\sigma$-codimensional Ahlfors regular with respect to $\nu$ (see Definition def:codim-regular), then there exist bounded linear trace and extension operators such that $T_Z\circ E_Z$ is the identity map on $B^{\theta-\sigma/p}_{p,p}(\partial Z,\pi)$. For ea

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.4
  • Lemma 2.6
  • proof
  • Theorem 2.7
  • Definition 2.8
  • Lemma 2.9
  • ...and 38 more