On the asymptotic Dirichlet problem for a class of mean curvature type partial differential equations
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Calculus of Variations
On the asymptotic Dirichlet problem for a class of mean curvature type partial differential equations Leonardo Bonorino1 · Jean-Baptiste Casteras2 · Patricia Klaser3 · Jaime Ripoll1,3 · Miriam Telichevesky1 Received: 20 December 2018 / Accepted: 31 May 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We study the Dirichlet problem for the following prescribed mean curvature PDE ⎧ ⎨− div ∇v = f (x, v) in 1 + |∇v|2 ⎩ v = ϕ on ∂, where is a domain contained in a complete Riemannian manifold M, f : × R → R is a fixed function and ϕ is a given continuous function on ∂. This is done in three parts. In the first one we consider this problem in the most general form, proving the existence of solutions when is a bounded C 2,α domain, under suitable conditions on f , with no restrictions on M besides completeness. In the second part we study the asymptotic Dirichlet problem when M is the hyperbolic space Hn and is the whole space. This part uses in an essential way the geometric structure of Hn to construct special barriers which resemble the Scherk type solutions of the minimal surface PDE. In the third part one uses these Scherk type graphs to prove the non existence of isolated asymptotic boundary singularities for global solutions of this Dirichlet problem. Mathematics Subject Classification 35J93 · 58J05 · 58J32
1 Introduction A natural way of finding bounded entire solutions to a partial differential equation on a Cartan– Hadamard manifold (complete, simply connected Riemannian manifold with nonpositive sectional curvature) is by solving the asymptotic Dirichlet problem with a prescribed boundary data given at infinity. This problem has been extensively studied for the Laplace equation mostly motivated by the Green–Wu conjecture which asserts the existence of bounded non constant harmonic functions on a Cartan–Hadamard manifold under certain growth and decay
Communicated by L. Ambrosio.
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Jaime Ripoll [email protected]
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conditions on the sectional curvature (see [9,13]). In the last years the asymptotic Dirichlet problem has been studied for other partial differential equations such as the p-Laplace ([10]) and the minimal surface equation ([4,5,7,12]). In our paper we study the Dirichlet problem for the following prescribed mean curvature PDE ⎧ ⎨− div ∇v = f (x, v) in (1.1) 1 + |∇v|2 ⎩ v = ϕ on ∂, where is a domain contained in a complete Riemannian manifold M, f : × R → R is a fixed function satisfying some conditions and ϕ is a given continuous function on ∂. The objective of this paper is threefold: first, to investigate the existence of solutions of (1.1) when is bounded; second, to study the asymptotic Dirichlet problem in the case where M is the hyperbolic space Hn and, third, to study the existence or not of isolated asymptotic boundary singularities for the solutions to the problem discussed
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