Conformal composition operators with applications to Dirichlet eigenvalues

This paper is concerned with spectral estimates for the first Dirichlet eigenvalue of the degenerate $p$-Laplace operator in bounded simply connected domains $Ω\subset \mathbb C$. The proposed approach relies on the conformal analysis of the elliptic operators, which allows us to obtain spectral estimates in domains with non-rectifiable boundaries.

💡 Research Summary

The paper addresses the problem of estimating the first Dirichlet eigenvalue of the degenerate p‑Laplacian (p>2) on bounded simply‑connected planar domains Ω⊂ℂ. The authors develop a novel approach that combines conformal geometry, composition operators on Sobolev spaces, and weighted Sobolev–Poincaré inequalities to produce both upper and lower bounds for λ₁^{(p)}(Ω).

First, the authors recall that the first eigenvalue can be characterized by the Rayleigh quotient
λ₁^{(p)}(Ω)=inf_{f∈W₀^{1,p}(Ω){0}} ∫Ω|∇f|^{p} / ∫Ω|f|^{p},
and that the reciprocal λ₁^{(p)}(Ω)^{-1} coincides with the optimal constant A{p,p}(Ω) in the Sobolev–Poincaré inequality ∥f∥{L^{p}(Ω)} ≤ A_{p,p}(Ω)∥∇f∥_{L^{p}(Ω)}. Classical estimates such as Rayleigh‑Faber‑Krahn, Cheeger, and inradius bounds are mentioned, but they do not capture the fine geometry of domains with irregular or fractal boundaries.

The core of the paper is the introduction of “conformal α‑regular domains”. A domain Ω is called conformal α‑regular with respect to a reference domain 𝔈 if there exists a conformal map φ:𝔈→Ω whose Jacobian J(w,φ) belongs to L^{α}(𝔈) for some α>1. This condition is independent of the particular choice of φ and can be expressed in terms of conformal radii. It includes Lipschitz domains, domains with Hölder continuous boundaries, and many fractal examples (e.g., snowflake‑type domains) whose boundary Hausdorff dimension lies in (1,2).

Next, the authors study composition operators induced by diffeomorphisms. Theorem 2.1 provides a necessary and sufficient condition for the boundedness of φ*:L^{p}(Ω)→L^{q}(𝔈) in terms of an integral involving |Dφ|^{p} and |J|^{q‑p}. For conformal maps, the condition simplifies dramatically: for any p>2 and q∈