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Robust nonlocal trace and extension theorems

Florian Grube, Moritz Kassmann

TL;DR

The paper constructs a robust trace/extension framework for nonlocal Sobolev-type spaces $V^{s,p}(\Omega|\mathds{R}^d)$, with exterior trace data on $\Omega^c$ encoded in the space $\mathcal{T}^{s,p}(\Omega^c)$ and the boundary-adapted measure $\mu_s$. It proves that the trace operator $\mathrm{Tr_s}$ is linear and continuous and possesses a continuous right inverse $\mathrm{Ext_s}$, establishing a full variational setup for nonlinear nonlocal problems such as the fractional $p$-Laplacian and ensuring robustness as $s\to1^{-}$, including $p=1$ via BV theory and Hardy inequalities. The authors develop a Whitney-decomposition based extension construction with explicit operator-norm bounds, and show that trace spaces converge to classical boundary traces on $\partial\Omega$ in the local limit, recovering $W^{1-1/p,p}(\partial\Omega)$ (or $BV$-Besov limits) in the appropriate regimes. These results enable well-posed nonlocal Dirichlet problems with exterior data and provide a principled link between nonlocal models and their local counterparts in Lipschitz domains, with potential impact on peridynamics and nonlocal PDE analysis.

Abstract

We prove trace and extension results for Sobolev-type function spaces that are well suited for nonlocal Dirichlet and Neumann problems including those for the fractional $p$-Laplacian. Our results are robust with respect to the order of differentiability. In this sense they are in align with the classical trace and extension theorems.

Robust nonlocal trace and extension theorems

TL;DR

The paper constructs a robust trace/extension framework for nonlocal Sobolev-type spaces , with exterior trace data on encoded in the space and the boundary-adapted measure . It proves that the trace operator is linear and continuous and possesses a continuous right inverse , establishing a full variational setup for nonlinear nonlocal problems such as the fractional -Laplacian and ensuring robustness as , including via BV theory and Hardy inequalities. The authors develop a Whitney-decomposition based extension construction with explicit operator-norm bounds, and show that trace spaces converge to classical boundary traces on in the local limit, recovering (or -Besov limits) in the appropriate regimes. These results enable well-posed nonlocal Dirichlet problems with exterior data and provide a principled link between nonlocal models and their local counterparts in Lipschitz domains, with potential impact on peridynamics and nonlocal PDE analysis.

Abstract

We prove trace and extension results for Sobolev-type function spaces that are well suited for nonlocal Dirichlet and Neumann problems including those for the fractional -Laplacian. Our results are robust with respect to the order of differentiability. In this sense they are in align with the classical trace and extension theorems.
Paper Structure (9 sections, 25 theorems, 202 equations)

This paper contains 9 sections, 25 theorems, 202 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Function spaces
  5. Trace Theorem
  6. Extension results
  7. Nonlocal to Local
  8. Reflected Whitney cubes
  9. Hardy inequality for the half space

Key Result

theorem 2

Let $\Omega\subset \mathds{R}^d$ be a bounded Lipschitz domain, $s\in(0,1)$, $1< p<\infty$. Then the trace operator is continuous and linear and there exists a continuous linear right inverse which we call the nonlocal extension operator. Moreover, the continuity constants of the linear trace and extension operator only depend on $\Omega$, a lower bound on $s$ as well as a lower and upper bound

Theorems & Definitions (52)

  • remark 1
  • theorem 2
  • theorem 3
  • theorem 4
  • remark 5
  • definition 6
  • corollary 7
  • proof
  • remark 8
  • remark 9
  • ...and 42 more