[Paper Review] On Trace Theorems for Sobolev Spaces

Pier Domenico Lamberti, Luigi Provenzano|arXiv (Cornell University)|Oct 9, 2019

Advanced Mathematical Modeling in Engineering23 references4 citations

TL;DR

This paper presents a novel Fourier-based approach using multi-parameter polyharmonic Steklov problems to explicitly characterize the trace spaces of Sobolev functions $ H^k(Ω) $ on Lipschitz domains for $ k \geq 2 $, extending prior results for $ k=2 $ and providing a spectral decomposition of traces via eigenfunctions of boundary value problems. The key contribution is a systematic spectral-theoretic description of trace spaces that generalizes Auchmuty's $ H^1 $ framework to higher-order Sobolev spaces.

ABSTRACT

We survey a few trace theorems for Sobolev spaces on $N$-dimensional Euclidean domains. We include known results on linear subspaces, in particular hyperspaces, and smooth boundaries, as well as less known results for Lipschitz boundaries, including Besov's Theorem and other characterizations of traces on planar domains, polygons in particular, in the spirit of the work of P. Grisvard. Finally, we present a recent approach, originally developed by G. Auchmuty in the case of the Sobolev space $H^1(Ω)$ on a Lipschitz domain $Ω$, and which we have further developed for the trace spaces of $H^k(Ω)$, $k\geq 2$, by using Fourier expansions associated with the eigenfunctions of new multi-parameter polyharmonic Steklov problems.

Motivation & Objective

• To extend the spectral characterization of trace spaces from $ H^1(\Omega) $ to $ H^k(\Omega) $ for $ k \geq 2 $ on Lipschitz domains.
• To provide an explicit description of the trace spaces $ \gamma_j(W^{k,2}(\Omega)) $ and the total trace space $ \Gamma(W^{k,2}(\Omega)) $ using eigenfunctions of multi-parameter polyharmonic Steklov problems.
• To generalize Auchmuty's approach for $ H^1 $ to higher-order Sobolev spaces by constructing new boundary eigenvalue problems for the polyharmonic operator.
• To establish a connection between trace space structure and the spectral properties of boundary differential operators associated with $ (-\Delta)^k $.

Proposed method

• Constructing a family of multi-parameter Steklov-type boundary value problems for the polyharmonic operator $ \Delta^k $, parameterized by $ \lambda $ and $ \mu $, to generate eigenfunctions that span the trace space.
• Using weak formulations of the boundary value problems (e.g., (4.24) and (4.25)) to define eigenvalue problems with boundary operators involving normal derivatives and surface divergence of the Hessian.
• Employing Fourier expansions in terms of eigenfunctions of these Steklov problems to decompose traces of $ H^k(\Omega) $ functions into spectral components.
• Defining boundary differential operators $ \mathcal{N}_j $ of order $ j+k $ that generalize Neumann-type conditions for the polyharmonic operator.
• Deriving the asymptotic behavior and Weyl-type eigenvalue asymptotics for the eigenvalues $ \sigma^{(\ell)}(\beta^{(\ell)}_0, \dots, \beta^{(\ell)}_{k-1}) $, ensuring completeness of the spectral basis.
• Establishing that the eigenfunctions of these problems form a Riesz basis in the trace space, enabling a stable and explicit representation of traces.

Experimental results

Research questions

• RQ1 How can the trace space of $ H^k(\Omega) $ for $ k \geq 2 $ be explicitly characterized on a Lipschitz domain $ \Omega $, beyond the classical Besov space description?
• RQ2 What is the spectral structure of the trace space when $ k \geq 2 $, and how can it be linked to eigenfunctions of a boundary value problem for the polyharmonic operator?
• RQ3 Can the $ H^1 $-based approach of Auchmuty be generalized to higher-order Sobolev spaces via a family of multi-parameter Steklov problems?
• RQ4 What is the role of the boundary differential operators $ \mathcal{N}_j $ in characterizing the traces of $ H^k(\Omega) $ functions?
• RQ5 How do the eigenvalues $ \sigma^{(\ell)} $ of the constructed Steklov problems depend on the parameters $ \beta^{(\ell)}_j $, and what is their asymptotic behavior?

Key findings

• The trace space $ \gamma_0(H^k(\Omega)) $ for $ k \geq 2 $ on a Lipschitz domain $ \Omega $ is characterized via eigenfunctions of a multi-parameter Steklov problem for the polyharmonic operator $ \Delta^k $, generalizing Auchmuty's $ H^1 $ result.
• For $ k=2 $, the trace space is described by eigenfunctions of problems (4.24) and (4.25), which are shown to be limits of other problems as $ \lambda, \mu \to -\infty $, confirming consistency with known spectral limits.
• The eigenvalues $ \sigma^{(0)}_j(\lambda) $ and $ \sigma^{(1)}_j(\mu) $ of the constructed problems form increasing sequences and satisfy Weyl-type asymptotics, ensuring the existence of a complete spectral basis.
• The boundary differential operators $ \mathcal{N}_j $, which generalize Neumann conditions, are explicitly identified as $ \mathcal{N}_0 u = \frac{\partial^2 u}{\partial \nu^2} $ and $ \mathcal{N}_1 u = -\text{div}_{\partial\Omega}((D^2 u \cdot \nu)_{\partial\Omega}) - \frac{\partial \Delta u}{\partial \nu} $ for $ k=2 $.
• The method provides a complete and explicit spectral decomposition of the trace space via Fourier expansions in the eigenfunctions of the Steklov problems, enabling stable and constructive trace representations.
• The framework extends to arbitrary $ k \geq 2 $, with the general eigenvalue problem (4.13) yielding a family of boundary value problems whose eigenfunctions form a Riesz basis for the trace space of $ H^k(\Omega) $.