Behaviour at infinity for solutions of a mixed boundary value problem via inversion

1. Problem Statement

The paper investigates a mixed boundary value problem for a quasilinear elliptic equation in an unbounded domain. Specifically, the authors consider the equation:
$$\text{div } A(x, \nabla u(x)) = 0$$
defined in an open infinite circular half-cylinder $G = B' \times (0, \infty) \subset \mathbb{R}^n$, where $B'$ is the unit ball in $\mathbb{R}^{n-1}$.

The boundary conditions are mixed:

• Dirichlet condition: Prescribed continuous data $u = f$ on a closed subset $F \subset \partial G$.
• Neumann condition: Zero conormal derivative $N_u(x) := A(x, \nabla u) \cdot \nu(x) = 0$ on the remainder of the boundary $\partial G \setminus F$.

The vector field $A(x, \nabla u)$ satisfies standard structural conditions for $p$-elliptic operators ($p > 1$), including monotonicity, coercivity, and homogeneity. The $p$-Laplace equation is a specific case included in this framework.

The primary objectives are:

1. To prove the existence and uniqueness of bounded weak solutions with continuous Dirichlet data.
2. To characterize the regularity of the point at infinity (i.e., whether the solution attains the limit of the boundary data as $x \to \infty$).
3. To describe the asymptotic behavior of solutions when the Dirichlet boundary is bounded (i.e., near infinity, only Neumann conditions apply), establishing a Phragmén–Lindelöf trichotomy.

2. Methodology

The core innovation of the paper is the use of a geometric inversion (change of variables) to transform the problem from an unbounded domain to a bounded one, allowing the application of established theories for degenerate elliptic equations.

A. The Inversion Transformation

The authors employ a mapping $T: G \to D$, where $D$ is the open unit upper half-ball (with the origin removed). The mapping is defined as:
$$\xi' = \frac{2e^{-\kappa x_n} x'}{1 + |x'|^2}, \quad \xi_n = \frac{e^{-\kappa x_n}(1 - |x'|^2)}{1 + |x'|^2}$$
Key properties of this transformation:

• The point at infinity in the cylinder ($x_n \to \infty$) maps to the origin ($\xi = 0$) in the ball.
• The quasilinear operator $\text{div } A(x, \nabla u)$ transforms into a weighted degenerate elliptic operator $\text{div } B(\xi, \nabla \tilde{u}) = 0$ in the unit ball.
• The weight function associated with the transformed operator is $w(\xi) = |\xi|^{p-n}$, which belongs to the Muckenhoupt $A_p$ class.

B. Elimination of Neumann Data via Reflection

A critical step in the methodology is handling the mixed boundary conditions.

• The Neumann condition (zero conormal derivative) on the flat part of the boundary corresponds to a symmetry condition in the transformed domain.
• The authors extend the domain $D$ to the full unit ball by reflecting across the hyperplane $\xi_n = 0$.
• By defining the operator $B$ symmetrically in the reflected half-ball, the Neumann condition is automatically satisfied. This reduces the mixed problem in the cylinder to a pure Dirichlet problem in the full unit ball with a degenerate weight.

C. Application of Existing Theory

Once reduced to a Dirichlet problem with an $A_p$ weight, the authors utilize tools from the theory of degenerate elliptic equations, specifically referencing Heinonen–Kilpeläinen–Martio [10] and Björn–Björn–Mwasa [2]. They apply:

• Existence and uniqueness theorems for Perron solutions.
• Wiener criteria for boundary regularity.
• Capacity estimates and Harnack inequalities.

3. Key Contributions and Results

A. Existence and Uniqueness (Theorem 5.3)

The authors prove that for continuous Dirichlet data $f$ on a closed set $F$ (which may be unbounded), if the limit $f(\infty)$ exists and is finite, there exists a unique bounded continuous weak solution $u$.

• The solution attains the boundary data $f(x_0)$ for all $x_0 \in F$ except possibly on a set of $C_p$-capacity zero.
• The solution is Hölder continuous at all points on the Neumann boundary.

B. Regularity at Infinity (Theorem 6.5)

The paper characterizes the regularity of the point at infinity using a Wiener-type criterion.

• Result: The point at infinity is regular (i.e., $\lim_{x \to \infty} u(x) = f(\infty)$) if and only if the Dirichlet boundary $F$ is "sufficiently thick" at infinity.
• Criterion: This thickness is measured by the divergence of an integral involving the variational $p$-capacity of the intersection of $F$ with truncated cylinders:
  $$\int_1^\infty \text{cap}_{p, G_{t-1}}(F \cap (G_t \setminus G_{2t}))^{1/(p-1)} \, dt = \infty$$
• If $F$ is bounded (meaning the Neumann condition holds near infinity), the point at infinity is irregular.

C. Phragmén–Lindelöf Trichotomy (Theorem 7.4)

When the Dirichlet boundary $F$ is bounded (so the Neumann condition applies near infinity), the authors establish a trichotomy for the asymptotic behavior of any solution $u$. As $x \to \infty$, the solution must behave in exactly one of three ways:

1. Convergence: The solution has a finite limit $u(\infty)$. The convergence is exponential: $|u(x) - u(\infty)| \lesssim e^{-\kappa \alpha x_n}$.
2. Linear Growth/Decay: The solution tends to $+\infty$ or $-\infty$ roughly linearly (i.e., $u(x) \sim A x_n + B$).
3. Oscillation: The solution changes sign and is unbounded in both directions, satisfying $\limsup_{x \to \infty} u(x) = \infty$ and $\liminf_{x \to \infty} u(x) = -\infty$.

This result generalizes known trichotomy results for linear elliptic equations to the quasilinear mixed setting.

4. Significance

• Extension to Quasilinear Equations: Previous results on mixed boundary value problems in unbounded domains were largely restricted to linear equations (e.g., Laplace or linear divergence form). This paper extends these results to general quasilinear operators, including the $p$-Laplacian.
• Unified Approach via Inversion: The use of the inversion transformation combined with reflection provides a powerful and elegant method to handle mixed boundary conditions in unbounded domains, converting them into standard Dirichlet problems on bounded domains with weights.
• Sharp Characterization: The paper provides a precise Wiener criterion for regularity at infinity, linking the geometric "thickness" of the boundary set to the analytic behavior of the solution.
• Trichotomy for Mixed Problems: The establishment of the Phragmén–Lindelöf trichotomy for mixed problems fills a gap in the literature, showing that even with mixed boundary conditions, solutions exhibit a structured set of asymptotic behaviors similar to pure Neumann or Dirichlet cases.

In summary, the paper successfully bridges the gap between the theory of degenerate elliptic equations on bounded domains and mixed boundary value problems on unbounded cylindrical domains, providing rigorous existence, uniqueness, and asymptotic classification results.