Abstract
We investigate overdetermined problems for p-Laplace and generalized Monge–Ampére equations. By using the theory of domain derivative, we find duality results and characterization of the overdetermined boundary conditions via minimization of suitable functionals with respect to the domain.
Highlights
Let D be a bounded smooth domain in RN
In Subsection 3.1 we prove a duality result for a p-Laplace boundary value problem
In Subsection 3.2 we prove a duality result for a boundary value problem corresponding to a generalized MongeAmpere equation
Summary
∂D holds for all functions v harmonic in D Motivated by this result, we shall prove duality theorems for overdetermined problems involving p-Laplace equations as well as generalized Monge-Ampere equations. In case of generalized Monge-Ampere equations, the overdetermined boundary condition is not the same as (2), but condition (27) below. We consider suitable functionals of the domain D whose minimizers must satisfy the overdetermined boundary condition (2) for the p-Laplace problem, and condition (27) for the generalized Monge-Ampere problems. In Subsection 3.1 we prove a duality result for a p-Laplace boundary value problem (see Theorem 3.1). In Subsection 3.2 we prove a duality result for a boundary value problem corresponding to a generalized MongeAmpere equation (see Theorem 3.2).
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