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Fractional $p$-Laplacians via Neumann problems in unbounded metric measure spaces

Luca Capogna, Ryan Gibara, Riikka Korte, Nageswari Shanmugalingam

Abstract

We prove well-posedness, Harnack inequality and sharp regularity of solutions to a fractional $p$-Laplace non-homogeneous equation $(-Δ_p)^su =f$, with $0<s<1$, $1<p<\infty$, for data $f$ satisfying a weighted $L^{p'}$ condition in a doubling metric measure space $(Z,d_Z,ν)$ that is possibly unbounded. Our approach is inspired by the work of Caffarelli and Silvestre \cite{CS} (see also Mol{č}anov and Ostrovski{ĭ} \cite{MO}), and extends the techniques developed in \cite{CKKSS}, where the bounded case is studied. Unlike in \cite{EbGKSS}, we do not assume that $Z$ supports a Poincaré inequality. The proof is based on the well-posedness of the Neumann problem on a Gromov hyperbolic space $(X,d_X, μ)$ that arises as an hyperbolic filling of $Z$.

Fractional $p$-Laplacians via Neumann problems in unbounded metric measure spaces

Abstract

We prove well-posedness, Harnack inequality and sharp regularity of solutions to a fractional -Laplace non-homogeneous equation , with , , for data satisfying a weighted condition in a doubling metric measure space that is possibly unbounded. Our approach is inspired by the work of Caffarelli and Silvestre \cite{CS} (see also Mol{č}anov and Ostrovski{ĭ} \cite{MO}), and extends the techniques developed in \cite{CKKSS}, where the bounded case is studied. Unlike in \cite{EbGKSS}, we do not assume that supports a Poincaré inequality. The proof is based on the well-posedness of the Neumann problem on a Gromov hyperbolic space that arises as an hyperbolic filling of .

Paper Structure

This paper contains 28 sections, 32 theorems, 180 equations.

Table of Contents

  1. Introduction
  2. Strategy of the proof
  3. Background definitions and results
  4. Doubling property and codimensionality
  5. Sobolev-type spaces and Poincaré inequalities
  6. Besov spaces
  7. Trace and extension theorems
  8. Uniform domains
  9. Differentiable structures
  10. Conformal invariance of $p$-harmonicity
  11. An equivalent formulation of the Neumann problem
  12. Dirichlet problem in unbounded domains with bounded boundary
  13. Uniform boundedness principle
  14. First Step: Dirichlet problem on domains with unbounded boundary
  15. Dirichlet problem on $\Omega_k$
  16. ...and 13 more sections

Key Result

Theorem 1.4

Assume hypotheses (H0), (H1) and (H2) above, and consider data with $\int_{\partial \Omega} f d\nu=0$. Here, for a fixed $x_0\in\partial\Omega$, we have set and the measure $\nu_J$ is defined by $\nu_J(E)=\int_E J(y,x_0)d\nu(y)$ whenever $E\subset \partial \Omega$ is a $\nu$-measurable set. The following properties hold:

Theorems & Definitions (71)

  • Remark 1.2
  • Theorem 1.4
  • Remark 1.9
  • Remark 1.10
  • Theorem 1.11
  • Remark 1.13
  • Remark 1.14
  • Remark 1.15
  • Remark 1.17
  • Remark 1.18
  • ...and 61 more