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- Title
On the asymptotic Dirichlet problem for a class of mean curvature type partial differential equations.
- Authors
Bonorino, Leonardo; Casteras, Jean-Baptiste; Klaser, Patricia; Ripoll, Jaime; Telichevesky, Miriam
- Abstract
We study the Dirichlet problem for the following prescribed mean curvature PDE - div ∇ v 1 + | ∇ v | 2 = f (x , v) in Ω v = φ on ∂ Ω , where Ω is a domain contained in a complete Riemannian manifold M, f : Ω × R → R is a fixed function and φ is a given continuous function on ∂ Ω . 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 Ω 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 H n and Ω is the whole space. This part uses in an essential way the geometric structure of H n 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.
- Subjects
DIRICHLET problem; PARTIAL differential equations; CURVATURE; HYPERBOLIC spaces; MINIMAL surfaces; RIEMANNIAN manifolds; CONTINUOUS functions
- Publication
Calculus of Variations & Partial Differential Equations, 2020, Vol 59, Issue 4, p1
- ISSN
0944-2669
- Publication type
Article
- DOI
10.1007/s00526-020-01795-5