Outward Cuspidal Domains in Geometric Analysis

• Outward cuspidal domains are non-Lipschitz regions defined by a narrowing cusp controlled by an increasing function, modeling singular boundary behavior.
• They reveal the equivalence of Sobolev and Hajłasz–Sobolev spaces, enabling compact weighted trace embeddings and bi-Lipschitz extension methods.
• These domains serve as model settings in spectral theory, facilitating analysis of Steklov, Robin, and Neumann eigenvalue problems under weighted conditions.

An outward cuspidal domain is a class of non-Lipschitz domains in $\mathbb{R}^n$ whose boundary possesses a singular “cusp” that narrows toward a tip or singular point, typically described by a continuous or left-continuous, strictly increasing function $\psi$ or $\phi$ controlling the cross-sectional “width” as one approaches the tip. These domains arise in geometric analysis, PDE spectral theory, Sobolev space trace and extension theory, and metric measure theory, and they offer model settings for the breakdown and subtle repair of classical functional-analytic properties in the non-Lipschitz regime.

1. Precise Geometric Definitions and Model Constructions

Let $n \geq 2$ and let $\psi:(0,1]\to(0,\infty)$ be a left-continuous, increasing function, extended by $\psi(t) = \psi(1)$ for $t\in(1,2)$. Points in $\mathbb{R}^n$ are denoted $z = (t,x)$, with $t\in\mathbb{R}$, $x\in\mathbb{R}^{n-1}$. The outward cuspidal domain is defined by

$$\Omega_\psi := \{ (t, x) \in (0,2)\times\mathbb{R}^{n-1} : |x| < \psi(t) \}.$$

Equivalently, near $t=0$ the domain has a “cusp” that narrows as $t\downarrow 0$ whenever $\psi(t)/t \to 0$, with boundary described by $|x| = \psi(t)$ (Eriksson-Bique et al., 2019). More generalizations allow star-shaped cross-sections, $\Sigma \subset \mathbb{R}^{n-1}$, so that

$$\Omega_\psi = \{ (t,x) : t \in (0,2), \; (x-x_0)/\psi(t) + x_0 \in \Sigma \}$$

where $x_0$ is a star-center for $\Sigma$ (Zhu, 2021). This setting includes the classical “power-cusps” $\psi(t)=t^s$ with $s>1$.

In spectral geometry, the “outward peak of order $q>1$” is defined locally by

$$\Omega \cap \{ 0 < x_1 < \delta \} = \{ (s, s^q t) : 0 < s < \delta, \; t \in S \}$$

where $S \subset \mathbb{R}^{d-1}$ is a bounded Lipschitz domain (Pankrashkin et al., 7 Apr 2025).

2. Sobolev and Hajłasz–Sobolev Space Equivalence

Outward cuspidal domains generally fail the measure-density or Whitney extension property, precluding direct Sobolev extension in classical terms. However, for any left-continuous, increasing profile function $\psi$, Eriksson-Bique, Koskela, Malý, and Zhu demonstrate that the first-order Sobolev space $W^{1,p}(\Omega_\psi)$ coincides with the Hajłasz–Sobolev space $M^{1,p}(\Omega_\psi)$, via pointwise inequalities: $W^{1,p}(\Omega_\psi) = M^{1,p}(\Omega_\psi)$ with equivalent norms and explicit pointwise control: $$|u(z_1) - u(z_2)| \le |z_1 - z_2| (g(z_1) + g(z_2)), \quad g \in L^p(\Omega_\psi)$$ where $u \in W^{1,p}(\Omega_\psi) \cap C^1(\Omega_\psi)$ and $g$ is formed with maximal operator estimates on $|\nabla u|$ and derivatives of suitable $x$-extensions $\tilde u$ (Eriksson-Bique et al., 2019, Zhu, 2021). This result generalizes to star-shaped cross-sections and profiles $\psi$, showing no geometric restriction (other than star-shapedness of the base) is needed.

3. Weighted Trace Embeddings and Compactness

Weighted embedding theorems are central in handling the singularity at the cusp tip. For $$\Omega_\gamma = \{ (x', x_n) : 0 < x_n \le 1, |x'| < x_n^\alpha \} \cup \text{smooth cap}$$, and weight $w(x', x_n) = x_n^\alpha$ on the cuspidal boundary, the weighted trace operator

$$i : W^{1,p}(\Omega_\gamma) \to L^p_w(\partial\Omega_\gamma)$$

is compact for all $1 < p < \infty$ (Lamberti et al., 20 Jan 2026, Garain et al., 2024). The weight compensates for the vanishing cross-section near the cusp tip, restoring compactness and ensuring that the (weighted) Steklov spectra are discrete for arbitrary sharp cusps (all $\alpha > 1$).

For non-linear Neumann spectral problems, similar arguments show that $W^{1,p}(M_\gamma)$ embeds compactly into weighted $L^q$ spaces with model weights vanishing at the tip (Menovschikov et al., 25 Feb 2025).

4. Sobolev Extension Theory

While $\Omega_\psi$ generally does not allow classical Sobolev extension, Koskela–Zhu show any outward cuspidal domain is globally bi-Lipschitz equivalent to a Lipschitz cusp $\Omega_{\hat\psi}$ via explicit chart $\mathcal O$, allowing the full Maz'ya–Poborchi extension theory to transfer to arbitrary cusps (Koskela et al., 2021). The optimal $p,q$ ranges and sharp integral criteria on $\psi$ for extension are inherited—specifically, in power-cusps $\psi(t) = t^s$,

$p \ge \frac{(n-1)q}{n-1-q}$

is necessary and sufficient for extension into $W^{1,q}(\mathbb{R}^n)$. The flattening is performed piecewise, with uniform control of the Jacobian.

5. Spectral Theory: Steklov, Robin, and Neumann Problems

Weighted Steklov and Neumann eigenvalue problems on outward cuspidal domains are analyzed through compactness of trace embeddings and weighted variational formulations. For the nonlinear Steklov problem, the first non-trivial eigenvalue $\lambda_p$ is characterized by the infimum of the Rayleigh quotient under orthogonality constraints: $$\lambda_p = \inf_{u \neq 0,\, \int_{\partial\Omega} |u|^{p-2} u w = 0} \frac{\int_{\Omega} |\nabla u|^p}{\int_{\partial\Omega} |u|^p w}$$ and attained at a weak solution (Lamberti et al., 20 Jan 2026, Garain et al., 2024). The full spectrum is discrete and diverges according to the geometry and vanishing rate of $w$ near the cusp.

For Robin Laplacians in domains with peaks (cusps), the main asymptotic result as Robin parameter $\alpha \to \infty$ is

$$\lambda_j(R_\alpha) = -\alpha^2 + C_{j,q,S}\, \alpha^{2/(2+q)} + o(\alpha^{2/(2+q)})$$

where $C_{j,q,S}$ is determined by cross-sectional geometry via isoperimetric constants, and $L_q$ is a model Schrödinger operator (Pankrashkin et al., 7 Apr 2025).

6. Examples, Explicit Constructions, and Further Generalizations

• Classical power-cusps: $$\Omega = \{ (x', x_n) : |x'| < x_n^s, 0 < x_n < 1 \}$$ with $s>1$ serve as canonical examples (Eriksson-Bique et al., 2019).
• Star-shaped and iterated cusps: Domains built by dilating star-shaped cross-sections, iteratively adding cusp geometries in different coordinate directions, all inherit Sobolev-Hajłasz equivalence.
• Hilbert-Blumenthal cusp domains: In arithmetic geometry, outward cuspidal regions appear as fundamental domains for cusp stabilizers, with cross-section foliated by Dirichlet polytopes and equipped with left-invariant Sol metrics (Quinn et al., 2017).

Model	Boundary Profile ($\psi$, $\phi$)	Main Spectral Feature
Power-cusp	$t^s$ ($s>1$)	$W^{1,p} = M^{1,p}$ for all $p$, compact weighted trace
General star-cusp	Any left-continuous, incr. $\psi$	Sobolev–Hajłasz equivalence iff base $\Sigma$ shares it
Outward peak	$s^q t$	Robin spectral asymptotics controlled by $q$ and $S$

7. Directions: Regularity, Open Problems, and Applications

Current investigations address the full characterization of necessary and sufficient integral conditions on $\psi$ for Sobolev extension, relaxation of doubling or monotonicity in $\psi$, and the development of explicit Weyl-law formulae for eigenvalue growth (Koskela et al., 2021, Garain et al., 2024). Applications span PDE spectral theory, metric measure geometry, and analytic function theory for domains with prescribed cusp patterns (Gandhi et al., 2021).

Regularity near the cusp is regulated by weighted trace compactness; eigenfunctions are smooth in the interior and admit traces that decay at the singular tip. Singular domains such as these model the breakdown of classical PDE phenomena and the crucial role of weights or geometric invariants in restoring compactness and spectral discreteness.