The overdetermined Cauchy problem

A Class of Overdetermined Systems Defined by Tableaux: Involutiveness and Cauchy Problem

This article addresses the question of involutiveness and discusses the initial value problem for a class of overdetermined systems of partial differential equations which arise in the theory of integrable systems and are defined by tableaux.

The Cauchy Problem for Membranes

We show existence and uniqueness for timelike minimal submanifolds (world volume of p-branes) in ambient Lorentz manifolds admitting a time function in a neighborhood of the initial submanifold. The initial value formulation introduced and the conditions imposed on the initial data are given in purely geometric terms. Only an initial direction must be prescribed in order to provide uniqueness f...

Inhomogeneous Two-Parameter Abstract Cauchy Problem

An Overdetermined Problem for an Elliptic Equation

We consider the following overdetermined boundary value problem: ∆u+ λu+ μ = 0 in Ω, u = 0 on ∂Ω and ∂u/∂n = c on ∂Ω, where c 6= 0, λ and μ are real constants and Ω ⊂ R is a smooth bounded convex open set. We first show that it may happen that the problem has no solution. Then we study the existence of solutions for a wide class of domains. 2010 Mathematics Subject Classification: 35J05, 35R30.

A Note on Some Overdetermined Elliptic Problem

Ω 7−→ λ1(Ω) , under the volume constraint Vol(Ω) = α (where α ∈ (0,Vol(M)) is fixed) are called extremal domains. Smooth extremal domains are characterized by the property that the eigenfunctions associated to the first eigenvalue of the Laplace-Beltrami operator have constant Neumann boundary data [2]. In other words, a smooth domain is extremal if and only if there exists a positive function ...