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Asymptotic dirichlet problem for A-harmonic and minimal graph equations in Cartan-Hadamard manifolds

par Casteras, Jean-Baptiste ;Holopainen, Ilkka;Ripoll, Jaime Bruck
Référence Communications in analysis and geometry, 27, 4, page (809-855)
Publication Publié, 2019

Article révisé par les pairs

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Résumé :

We study the asymptotic Dirichlet problem for A-harmonic equations and for the minimal graph equation on a Cartan-Hadamard manifold M whose sectional curvatures are bounded from below and above by certain functions depending on the distance r = d(·, o) to a fixed point o ∈ M. We are, in particular, interested in finding optimal (or close to optimal) curvature upper bounds. In the special case of the Laplace-Beltrami equation we are able to solve the asymptotic Dirichlet problem in dimensions n ≥ 3 if radial sectional curvatures satisfy (Equation presented) outside a compact set for some ϵ > ϵ̄ > 0. The upper bound is close to optimal since the nonsolvability is known if (Equation presented) Our results (in the non-rotationally symmetric case) improve on the previously known case of the quadratically decaying upper bound.