Asymptotic Dirichlet Problem for 𝓐 $\mathcal {A}$ -Harmonic Functions on Manifolds with Pinched Curvature

Abstract

Abstract. We study the asymptotic Dirichlet problem for A-harmonic functions on a Cartan-Hadamard manifold whose radial sectional curvatures outside a compact set satisfy an upper boundand a pointwise pinching conditionfor some constants ε > 0 and C K ≥ 1, where P and P are any 2-dimensional subspaces of TxM containing the (radial) vector ∇r(x) and r(x) = d(o, x) is the distance to a fixed point o ∈ M . We solve the asymptotic Dirichlet problem with any continuous boundary data f ∈ C(∂∞M ).

On solvability of the boundary value problems for the inhomogeneous elliptic equations on noncompact Riemannian manifolds

“…However, the class of manifolds admitting nontrivial solutions of some elliptic equations is wide. For example, conditions ensuring the solvability of the Dirichlet problem with continuous boundary conditions "at infinity" for several noncompact manifolds has been found in many papers (see, e. g., [1,5,7,10,15]).…”

Existence and non-existence of minimal graphic and p-harmonic functions

“…Recently the asymptotic Dirichlet problem for minimal graph, f -minimal graph, p-harmonic and A-harmonic equations has been studied for example in [22], [30], [31], [7], [21], [8], [6], [5], and [17], where the existence of solutions was studied under various curvature assumptions and via different methods. In [8] the existence of solutions to the minimal graph equation and to the A-harmonic equation was proved in dimensions n ≥ 3 under curvature assumptions − log r(x)) 2ε r(x) 2 ≤ K(P x ) ≤ − 1 + ε r(x) 2 log r(x)…”

“…, (1.3) where ε >ε > 0, P x ⊂ T x M is a 2-dimensional subspace, x ∈ M \ B(o, R 0 ), and r(x) = d(o, x) is the distance to a fixed point o ∈ M . In [17] it was shown that in the case of A-harmonic functions the curvature lower bound can be replaced by a so-called pinching condition…”