Compactness of weighted Sobolev trace operators and non-linear Steklov problems

Abstract: We prove the compactness of weighted Sobolev trace operators in outward cuspidal domains by using composition operators on Sobolev spaces. This result allows us to formulate the non-linear Steklov problem in outward cuspidal domains in a correct functional setting and to establish the existence of its non-trivial solution.

• The paper demonstrates that the weighted Sobolev trace operator is compact on outward cuspidal domains by employing composition operators on Sobolev spaces.
• It rigorously establishes necessary and sufficient conditions, including sharp power weight bounds, to recover compact trace embeddings in singular settings.
• The study applies these compactness findings to prove existence and provide explicit bounds for non-linear (p,q)-type Steklov eigenvalue problems.

Compactness of Weighted Sobolev Trace Operators and Non-linear Steklov Problems

Introduction and Context

The analyzed work provides an in-depth investigation into the compactness of weighted Sobolev trace operators in outward cuspidal domains in $\mathbb{R}^n$, leveraging the machinery of composition operators on Sobolev spaces. This compactness result paves the way for the correct formulation and the existence theory for non-linear (Schrödinger-)Steklov $(p,q)$-type eigenvalue problems on such singular domains.

Outward cuspidal domains, defined by the mapping $g(t)=t^{\alpha}$ where $\alpha=(\gamma-1)/(n-1)$, $n < \gamma < \infty$, feature sharp geometric singularities at the boundary, which deeply affect the analytic and spectral properties of the associated PDEs. Classical Sobolev space theory and trace theorems, which are crucial for the analysis of boundary value problems, often fail or dramatically change in non-Lipschitz domains such as cusps. Thus, this paper addresses a long-standing issue regarding the identification of suitable weighted trace spaces where compactness can be recovered, which is a prerequisite for the application of variational methods to nonlinear spectral problems.

Compactness of Weighted Trace Operators

A principal contribution is the establishment of necessary and sufficient conditions for the compactness of the Sobolev trace operator

$$T: W^{1,p}(\Omega_\gamma) \to L^q(\partial\Omega_\gamma, w_\gamma),$$

where $\Omega_\gamma\subset\mathbb{R}^n$ is an outward $\gamma$-cuspidal domain, $1 < p < n$, $1 < q < \frac{p(n-1)}{n-p}$, and $(p,q)$0 is a boundary weight tailored to the local geometry of the cusp. Specifically, the sharp weight is

$(p,q)$1

This weight is derived from the tangential Jacobian of the cusp-straightening mapping, and precisely compensates for the anisotropic distortion of the boundary measure induced by the cusp.

The proof strategically utilizes the geometric theory of composition operators on Sobolev spaces. In particular, the operator is studied by pulling back the trace embedding from the singular domain $(p,q)$2 to a model Lipschitz domain $(p,q)$3 via carefully constructed homeomorphisms. The subsequent analysis of the composition operator ensures boundedness and compactness for the trace operator under explicit relationships between the parameters $(p,q)$4, $(p,q)$5, and $(p,q)$6.

A comprehensive necessary condition is also derived for power weights, showing that the weighted compact trace embedding is sharp within the class of power weights. The results substantially refine earlier trace embedding theorems for cuspidal domains (e.g., [GV, GGU25]) and provide explicit bounds and asymptotics for the involved weights.

Moreover, the framework includes a characterization of when unweighted trace embeddings are compact, introducing the concept of an effective boundary dimension dictated by the external cusp, and deriving the optimal range for the exponents $(p,q)$7 and $(p,q)$8.

Non-Linear Weighted Steklov Problems

Utilizing the compactness of the trace operator, the study then rigorously formulates the non-reduced weighted Schrödinger–Steklov $(p,q)$9-eigenvalue problem in outward cuspidal domains:

$g(t)=t^{\alpha}$0

The weak formulation is addressed as an Euler-Lagrange equation corresponding to the minimization of the Rayleigh quotient:

$g(t)=t^{\alpha}$1

Existence of a non-trivial eigenfunction is established for all $g(t)=t^{\alpha}$2 and $g(t)=t^{\alpha}$3, with the infimum in the variational characterization being achieved. The results also yield explicit lower bounds for the first non-trivial Steklov eigenvalue on singular domains.

Theoretical and Practical Implications

The sharp identification of the trace space via composition operator methods has implications for the geometric analysis of PDEs on non-smooth domains, extending the field of applicability of spectral theory to settings with singular boundaries.

Practically, these results provide the functional analytical foundation for addressing spectral optimization problems, nonlinear boundary phenomena, and the study of nonlinear PDEs under singular geometric constraints. In particular, they provide tools to analyze eigenvalue problems for nonlinear operators on domains arising in physics and materials science, where non-smooth geometries are natural.

Theoretically, these methods provide a template for analyzing affine-invariant inequalities and embedding theorems in weighted and singular settings, which may impact related fields such as quasiconformal analysis, potential theory, and the theory of weighted capacities. The methodology based on composition operators opens avenues for further refinements in more general metric measure spaces and for different classes of differential operators.

Furthermore, estimating optimal trace constants in terms of the cusp parameter $g(t)=t^{\alpha}$4 elucidates the precise effect of the geometric singularity on spectral quantities, and could inform future research on spectral asymptotics for nonlinear Steklov problems and trace inequalities.

Conclusion

This paper develops a comprehensive analytic framework for the study of compactness of weighted trace operators in outward cuspidal domains, identifies sharp weights ensuring compact embeddings, and uses these results to rigorously treat the existence theory for nonlinear Steklov-type eigenvalue problems in singular domains. The composition operator technique and the explicit quantification of the involved constants provide a robust, widely applicable methodology for the geometric analysis of PDEs on non-Lipschitz domains, and serve as a touchstone for future work on nonlinear boundary spectral problems in singular or weighted settings.