Boundary Regularity at Infinity

• Boundary regularity at infinity is the study of the limiting behavior of PDE solutions as the spatial variable approaches infinity, ensuring continuity and convergence of solutions.
• Methodologies include capacity-based Wiener integrals, barrier function constructions, and inversion techniques that transform unbounded domains into bounded ones for tractable analysis.
• The concept applies across various settings—from Euclidean to hyperbolic spaces and fractional operators—impacting both theoretical insights and practical applications in physics and geometry.

Boundary regularity at infinity refers to the precise behavior and continuity properties of solutions to PDEs – notably nonlinear elliptic and degenerate equations – as the spatial variable tends toward the "point at infinity" in unbounded domains. This concept is fundamental in both Euclidean and non-Euclidean geometries, as well as for a broad class of operators, including (weighted) $p$-Laplace type, fractional Laplacians, reaction-diffusion equations, and in physical applications (e.g., scalar wave equations in Minkowski or hyperbolic spaces). Regularity at infinity is typically characterized via capacity-based Wiener-type integrals, barrier function constructions, and is deeply linked to quantitative geometric and topological properties of the domain – such as the size or structure of the boundary at infinity – as well as the analytic properties of the underlying operator.

1. Definitions: Boundary Regularity at Infinity

Let $\Omega\subset\mathbb{R}^n$ (or in a metric space, manifold, or more generally, a geometric background) be an unbounded domain and $1<p < \infty$. The boundary point at infinity, denoted $\boldsymbol{\infty}$, is added to the one-point compactification of $X$ so that $\partial\Omega$ in $X^* = X \cup \{\boldsymbol{\infty}\}$ always contains infinity if $\Omega$ is unbounded (Björn et al., 2019).

A solution $u$ (e.g., $p$-harmonic, $p$-Laplace, minimal surface, etc.) to a boundary value problem on $\Omega$ with boundary data $f\in C(\partial\Omega\cup\{\boldsymbol{\infty}\})$ is said to be regular at infinity if

$$\lim_{x\to\boldsymbol{\infty}} u(x) = f(\boldsymbol{\infty})$$

for every continuous $f$ (Björn et al., 15 Nov 2025, Björn et al., 2019). In geometric contexts, such as Hadamard manifolds, regularity of a point $\xi\in\partial_\infty M$ for a quasilinear operator $Q$ is similarly defined in terms of solvability and continuity of boundary data up to $\xi$ (Ripoll et al., 2013).

Barrier regularity provides an equivalent characterization: $\boldsymbol{\infty}$ is regular iff there exists a superharmonic barrier at infinity vanishing there and bounded away from zero near finite boundary points (Björn et al., 2019).

2. Wiener-Type Capacity Criteria for Regularity at Infinity

The central analytic tool is a capacity-based ("Wiener-type") criterion linking the geometry of the domain at large scales to regularity at infinity. For nonlinear $p$-Laplace type equations ($p \ge n$), regularity is governed by the divergence of an integral involving the condenser $p$-capacity (Björn et al., 15 Nov 2025):

$$\int_{R_0}^\infty \left[ \frac{\mathrm{cap}_p( \Omega^c \cap B(0, r); B(0, 2r)) }{ r^{n-p} } \right]^{1/(p-1)} \frac{dr}{r} = \infty$$

for some (hence all) $R_0>0$.

For $p>n$, this criterion simplifies: infinity is regular if and only if the boundary $\partial\Omega$ is unbounded, as the capacity ratio stays bounded above and below whenever the complement is nonempty at arbitrarily large scales. For $p=n$, the logarithmic decay of annular capacity permits counter-examples: one may construct sets with $\partial\Omega$ unbounded but $\boldsymbol{\infty}$ irregular by sparseness (thinness) at infinity (Björn et al., 15 Nov 2025). For the case $p<n$, in Euclidean spaces classical Wiener tests take the form (Björn et al., 2019):

$$\int_R^\infty \left[ \frac{ \mathrm{cap}_p( E_t ; B(0, 2t)) }{ \mathrm{cap}_p( B(0, t); B(0, 2t)) } \right]^{1/(p-1)} \frac{dt}{t} = \infty$$

where $E_t$ is the part of the complement in $B(0, t)$.

In mixed boundary-value problems, similar Wiener-type integrals in terms of local (Neumann) capacities provide sharp criteria for regularity at infinity (Björn et al., 2021).

3. Circular Inversion and Sphericalization Techniques

For analytic tractability, the circular inversion map $I(x) = x/|x|^2$ is a crucial tool. Under $I$, the point at infinity is mapped to the origin (and vice versa), and unbounded domains are transformed into bounded ones containing zero. The original $p$-Laplace operator transforms into a weighted $p$-Laplace type equation, with the weight $w(\xi) = |\xi|^{2(p-n)}$, and capacities must be replaced with corresponding weighted capacities (Björn et al., 15 Nov 2025, Björn et al., 2021).

The invariance of Sobolev spaces and Perron solutions under inversion means regularity at infinity in the original domain is equivalent to regularity at the origin for the transformed weighted equation. Consequently, classical finite-boundary point criteria (such as the Wiener test) transfer directly to provide criteria at infinity.

This methodology extends to Ahlfors $Q$-regular metric measure spaces. Sphericalization replaces the metric and measure with conformally equivalent versions, rendering $\boldsymbol{\infty}$ a finite boundary point and enabling an analytic correspondence (Björn et al., 15 Nov 2025).

4. Regularity at Infinity: Geometric and Capacity-Based Dichotomies

The capacity-based framework yields sharp dichotomies:

• $p > n$ Regime:

The existence of arbitrarily large boundary components is sufficient (and necessary) for regularity at infinity. Domains with compact complement or compact boundary are always irregular at infinity.

• $p = n$ Regime:

The mere presence of unbounded boundary is insufficient; "thin" complements (e.g., closures of sequences of small balls) may be so sparse at infinity that the Wiener integral converges, yielding irregularity despite unboundedness (Björn et al., 15 Nov 2025).

• Mixed Boundary-Value Problems:

The Wiener-type test in terms of local Neumann capacity of the Dirichlet portion of the boundary leads to a Phragmén–Lindelöf trichotomy at infinity: boundedness and convergence, exact linear growth to $\pm\infty$, or sign-changing unbounded oscillation, depending only on the "thickness" of the boundary near infinity (Björn et al., 2021).

• Metric Space and Manifold Contexts:

In Ahlfors regular settings ($Q<p$), the same capacity-geometry correspondence applies; in Hadamard manifolds, strict convexity ("SC condition") ensures existence of barriers and hence regularity for a large class of elliptic operators (including $p$-Laplace and minimal surface equations) (Ripoll et al., 2013).

5. Barrier and Obstacle Problem Characterizations

The existence of a superharmonic barrier at infinity provides a functional analytic equivalent to the capacity criterion: such a barrier vanishes at infinity, stays positive near finite boundary points, and induces the correct limiting behavior for Perron solutions (Björn et al., 2019). The arguments for barrier construction are robust and local: regularity at infinity is determined solely by the geometry and capacity of the complement outside large balls.

This extends immediately to one-sided obstacle problems: if infinity is regular for the domain, then Perron-type regularized solutions to obstacle problems with boundary data converge at infinity if and only if the data and obstacle functions satisfy limiting inequalities as the point escapes to infinity (Björn et al., 2019).

6. Boundary Regularity at Infinity for Specific Operators and Geometries

Boundary regularity at infinity appears in many specialized contexts:

• Fractional Laplacians: For regional fractional Laplacians $(-\Delta)^s_\Omega$, under optimal regularity assumptions on $\Omega$ and with $f\in L^p$, precise $C^\alpha$-expansion is obtained near the boundary, with regularity up to infinity deduced via Liouville-type classifications and blow-up/compactness arguments transferring local half-space results to global boundary regularity (Fall, 2020).
• Infinity Laplacian: Geometric measure properties (e.g., finite Hausdorff dimension of the free boundary for plateau sets) and optimal $C^{1,1/3}$ regularity near the free boundary are demonstrated for obstacle and reaction-diffusion problems involving the infinity Laplacian (Rossi et al., 2012, Araújo et al., 2016).
• Constant Curvature Hypersurfaces in Hyperbolic Space: For vertical graphs over domains in hyperbolic space with prescribed asymptotic boundary, full boundary expansions reveal that $C^{p,\alpha}$ boundary data yields solutions of precisely $C^{p,\alpha}$ regularity up to infinity, unless a log-term obstruction vanishes. The absence of the obstruction guarantees $C^\infty$ regularity up to infinity (Jiang et al., 2018).
• Wave Equations at Null Infinity: In the relativistic context, characteristic data at past null infinity propagates regularity to future null infinity, with expansions controlled in terms of Sobolev norm of the initial data. Peeling behaviour is explicitly quantified, and absence of polyhomogeneity (logarithmic obstruction terms) is tied to analytic regularity of the initial data (Marajh et al., 6 Aug 2025).

7. Locality and Robustness of Regularity at Infinity

Regularity at infinity is a strictly local property: it depends only on the geometry and complement of the domain "near infinity" (i.e., outside large compact sets). Barriers, capacity integrals, and all analytic constructions can be restricted to arbitrarily large exterior regions without loss of generality (Björn et al., 2019). Consequently, regularity at infinity and associated results can be "patched" across domains and applied in highly abstract metric, geometric, or dynamical settings.

Table: Principal Boundary Regularity Criteria at Infinity

Operator/Class	Regularity Criterion at Infinity	Key Reference
$p$-Laplace ($p>n$, Euclidean)	$\boldsymbol{\infty}$ regular $\Leftrightarrow$ $\partial\Omega$ unbounded	(Björn et al., 15 Nov 2025)
$p$-Laplace ($p=n$)	Wiener integral divergence (log-scale); unbounded $\partial\Omega$ not sufficient	(Björn et al., 15 Nov 2025)
Ahlfors $Q$-regular metric space ($p>Q$)	$\boldsymbol{\infty}$ regular $\Leftrightarrow$ $\partial\Omega$ unbounded	(Björn et al., 15 Nov 2025)
Mixed quasilinear elliptic (half-cylinder)	Wiener-type integral in local Neumann capacity	(Björn et al., 2021)
Fractional Laplacian ($(-\Delta)^s_\Omega$)	$C^\alpha$ expansion near boundary at infinity	(Fall, 2020)
Hadamard manifolds, $p$-Laplace, minimal surface	Strict convexity (SC) ensures barriers, hence regularity	(Ripoll et al., 2013)
Wave equation (Minkowski spacetime)	Peeling controlled by Sobolev regularity of data	(Marajh et al., 6 Aug 2025)

• "Boundary regularity and Wiener-type criteria at infinity for nonlinear elliptic equations of $p$-Laplace type" (Björn et al., 15 Nov 2025)
• "Boundary regularity for $p$-harmonic functions and solutions of obstacle problems on unbounded sets in metric spaces" (Björn et al., 2019)
• "Behaviour at infinity for solutions of a mixed boundary value problem via inversion" (Björn et al., 2021)
• "Regularity at infinity of Hadamard manifolds with respect to some elliptic operators and applications to asymptotic Dirichlet problems" (Ripoll et al., 2013)
• "Regional fractional Laplacians: Boundary regularity" (Fall, 2020)
• "Optimal Regularity of Constant Graphs in Hyperbolic Space" (Jiang et al., 2018)
• "Controlled regularity at future null infinity from past asymptotic initial data: the wave equation" (Marajh et al., 6 Aug 2025)
• "Optimal regularity at the free boundary for the infinity obstacle problem" (Rossi et al., 2012)
• "Infinity Laplacian equation with strong absorptions" (Araújo et al., 2016)
• "On the boundary Hölder regularity for the infinity Laplace equation" (Wu et al., 2019)