[Paper Review] Function spaces and extension results for nonlocal Dirichlet problems

Moritz Kaßmann, Bartłomiej Dyda|arXiv (Cornell University)|Dec 6, 2016

Advanced Mathematical Modeling in Engineering4 citations

TL;DR

This paper introduces a new function space, denoted as the nonlocal trace space, to characterize boundary data for nonlocal Dirichlet problems involving integrodifferential operators. It establishes extension and trace theorems for functions defined on the complement of a domain, proving the existence of a bounded extension operator from the trace space to the solution space $ V^s(ar{\Omega}|\mathbb{R}^d) $, with sharp dependence on the fractional order $ s $, and extends classical Sobolev extension results in the limit $ s \to 1^- $.

ABSTRACT

We study function spaces and extension results in relation with Dirichlet problems involving integrodifferential operators. For such problems, data are prescribed on the complement of a given domain in the Euclidean space. We introduce a function space that serves as a trace space for nonlocal Dirichlet problems and study related extension results.

Motivation & Objective

• Address the lack of systematic function space theory for nonlocal Dirichlet problems where data are prescribed on the complement of a domain $ \Omega \subset \mathbb{R}^d $.
• Define a new function space that serves as a trace space for nonlocal problems, capturing the regularity of boundary data $ g $ on $ \Omega^c $.
• Establish an extension operator that maps functions $ g \in \text{trace space} $ to solutions $ u \in V^s(\Omega|\mathbb{R}^d) $ satisfying the nonlocal equation $ (-\Delta)^s u = 0 $ in $ \Omega $.
• Prove that the extension operator is bounded with operator norm controlled by $ \frac{1}{s(1-s)} $, revealing a sharp singularity as $ s \to 0 $ or $ s \to 1 $.
• Extend classical trace and extension results for Sobolev spaces by recovering them in the limit $ s \to 1^- $, providing a nonlocal analog to classical theory.

Proposed method

• Define the solution space $ V^s(\Omega|\mathbb{R}^d) $ as the set of functions in $ L^2_{\text{loc}}(\mathbb{R}^d) $ with finite Gagliardo-type seminorm involving differences over $ \mathbb{R}^d \times \mathbb{R}^d $, weighted by $ |x-y|^{-d-2s} $.
• Introduce the trace space as the restriction of $ V^s(\Omega|\mathbb{R}^d) $ to $ \Omega^c $, equipped with the induced seminorm, and define the extension operator via a partition of unity and reflection across dyadic cubes.
• Use a Whitney-type decomposition of $ \Omega $ and its complement to construct a localized extension operator that preserves local regularity and controls the nonlocal seminorm.
• Establish the key estimate $ |\operatorname{ext}(f)|_{\Omega^{\text{int}}_\delta, \Omega^{\text{ext}}_\varepsilon}^{s,p} \lesssim \frac{1}{s} |f|_{\Omega^{\text{ext}}_\delta, \Omega^{\text{ext}}_\varepsilon}^{s,p} $, showing the operator norm depends on $ 1/s $.
• Prove boundedness in weighted $ L^p $-spaces by comparing the size of cubes and their reflections, using the comparability $ |x|+1 \asymp |\tilde{x}|+1 $ for $ x \in Q $, $ \tilde{x} \in \widetilde{Q} $, with constants depending only on $ \Omega $.
• Derive the trace norm estimate $ \|\operatorname{ext}(f)\|_{V^s(\Omega|\mathbb{R}^d)} \lesssim \frac{1}{s(1-s)} \|f\|_{\text{trace space}} $, proving the extension operator is uniformly bounded as $ s \to 1^- $.

Experimental results

Research questions

• RQ1 What function space on $ \Omega^c $ serves as the correct trace space for nonlocal Dirichlet problems involving integrodifferential operators?
• RQ2 How can one construct a bounded linear extension operator from the trace space on $ \Omega^c $ to the solution space $ V^s(\Omega|\mathbb{R}^d) $?
• RQ3 What is the sharp dependence of the extension operator norm on the fractional order $ s \in (0,1) $, and how does it behave as $ s \to 1^- $?
• RQ4 Can classical extension results for Sobolev spaces be recovered as a limit of the nonlocal extension theory?
• RQ5 Is the extension operator bounded in weighted $ L^p $-spaces with weight $ (1+|x|)^\beta $, and what are the conditions on $ \beta $ and the domain?

Key findings

• The paper introduces a new trace space for nonlocal Dirichlet problems, defined as the restriction of $ V^s(\Omega|\mathbb{R}^d) $ to $ \Omega^c $, which characterizes the regularity of boundary data $ g $.
• The extension operator $ \operatorname{ext} $ maps functions $ f \in \text{trace space} $ to $ u \in V^s(\Omega|\mathbb{R}^d) $ such that $ u = g $ on $ \Omega^c $, with operator norm bounded by $ \frac{1}{s(1-s)} $.
• An explicit estimate $ |\operatorname{ext}(f)|_{\Omega^{\text{int}}_\delta, \Omega^{\text{ext}}_\varepsilon}^{s,p} \lesssim \frac{1}{s} |f|_{\Omega^{\text{ext}}_\delta, \Omega^{\text{ext}}_\varepsilon}^{s,p} $ is established, showing the operator norm blows up as $ s \to 0 $, but remains finite for $ s \in (0,1) $.
• The extension operator is bounded in weighted $ L^p $-spaces with weight $ (1+|x|)^\beta $, with the bound depending only on the domain $ \Omega $, not on $ f $, for $ p < \infty $.
• The limit $ s \to 1^- $ recovers classical extension results for Sobolev spaces, showing that the nonlocal theory generalizes and extends classical trace and extension theorems.
• An $ L^\infty $-boundedness result is proven: $ \|\operatorname{ext}(f)\|_{L^\infty(\Omega)} \lesssim \|f\|_{L^\infty(\Omega^c)} $, confirming pointwise control of the extension.