On the Dirichlet problem for a prescribed anisotropic mean curvature

On the Dirichlet problem
for a prescribed anisotropic mean curvature equation
Chiara Corsato
The seminar is focused on the existence, the uniqueness and
the regularity of solutions of the Dirichlet problem
!


b
−div p ∇u
= −au + p
in Ω,
1 + |∇u|2
1 + |∇u|2


u=0
on ∂Ω,
where a, b are positive parameters and Ω is a bounded domain
in RN with Lipschitz boundary. This problem has been recently
proposed as a model for the geometry of the human cornea. After a brief introduction about the derivation of the model, the
attention will be addressed to the solvability of the problem.
The method of proof relies on the degree theory in the onedimensional case and on the shooting method in the radial case
in a ball: for both problems classical solutions are provided.
The general N −dimensional case is treated in a non-classical
setting, by applying variational techniques to a suitable equivalent formulation of the problem. For complete details we refer
to [1, 2, 3].
References
[1] I. Coelho, C. Corsato and P. Omari, A one-dimensional prescribed curvature equation modeling the corneal shape, Boundary Value Problems
2014, 2014:127, 19 pp.
[2] C. Corsato, C. De Coster and P. Omari, Radially symmetric solutions
of an anisotropic mean curvature equation modeling the corneal shape.
Available at: Quaderno Matematico 642, Dipartimento di Matematica
e Geoscienze, Universit`
a degli Studi di Trieste (2014).
[3] C. Corsato, C. De Coster and P. Omari, The Dirichlet problem for a
prescribed anisotropic mean curvature equation: existence, uniqueness
and regularity of solutions, preprint.
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