On the nature of isolated asymptotic singularities of solutions of a family of quasi-linear elliptic PDE's on a Cartan–Hadamard manifold

Abstract

Let M be a Cartan-Hadamard manifold with sectional curvature satisfying −b 2 ≤ K ≤ −a 2 < 0, b ≥ a > 0. Denote by ∂ ∞ M the asymptotic boundary of M and byM := M ∪ ∂ ∞ M the geometric compactification of M with the cone topology. We investigate here the following question: Given a finite number of points..,p k } extends continuously to p i , i = 1, ..., k, can one conclude that u ∈ C 0 M ? When dim M = 2, for Q belonging to a linearly convex space of quasi-linear elliptic operators S of the form IntroductionLet M be Cartan-Hadamard n−dimensional manifold (complete, connected, simply connected Riemannian manifold with non-positive sectional curvature). It is well-known that M can be compactified with the so called cone topology by adding a sphere at infinity, also called the asymptotic

On the asymptotic Dirichlet problem for a class of mean curvature type partial differential equations

“…The proof of the next result follows the same idea as in Theorem 1.1 of [2]. According to Proposition 5.5, u is bounded from above by some M, so K ≤ M. We show now that, for any δ > 0, we have K ≤ δ.…”

“…Using the Scherk type solutions and following the same idea as in [2], we prove that a solution to this problem can be extended continuously to the points p i , that is, such a solution satisfies v = ϕ on ∂ ∞ H n . However, since our Scherk type solutions are not isometric, in contrast with the Scherk solutions used in [2], we need some auxiliary results to prove that the solutions are bounded. For that, remind that ψ = ψ S , defined in Proposition 3.2, satisfies Proof.…”

“…For q ∈ ∂ ∞ A ∩ ∂ ∞ (H n \A), let B q be a totally geodesic hyperball disjoint with V 2 such that q ∈ int∂ ∞ B q and B q ⊂ T j (V j ) for any j. (This is possible since (V j ) is a decreasing sequence, ∂T j (V j ) is a totally geodesic hypersphere, dist(∂T j (V j ), A) ≥ j as in [2] and, then some neighborhood of ∂ ∞ A ⊂ int ∂ ∞ T j (V j ) for any j). As in Theorem 1.2, consider the supersolutions…”

“…The existence or not of interior or boundary singularities for the minimal surface equation in the Euclidean space is a classical topic of study. In [2] the authors extended this study to the asymptotic Dirichlet problem in a Riemanian manifold and to a large class or partial differential equations which includes the p−Laplace and the minimal surface equation. In particular, they prove that isolated singularities at the asymptotic boundary of solutions of the minimal surface equation are removable.…”