Some nonstandard problems for the poisson equation

  title={Some nonstandard problems for the poisson equation},
  author={Lawrence Edward Payne and P. W. Schaefer},
  journal={Quarterly of Applied Mathematics},

Some fully nonlinear elliptic boundary value problems with ellipsoidal free boundaries

In this paper we study some overdetermined boundary value problems for three classes of fully nonlinear elliptic equations. In each case we prove that the solution exists if and only if the

The Isoperimetric Inequality via Approximation Theory and Free Boundary Problems

In this survey paper, we examine the isoperimetric inequality from an analytic point of view. We use as a point of departure the concept of analytic content in approximation theory: this approach

On an oblique boundary value problem related to the Backus problem in Geodesy

We show the existence and uniqueness of a viscosity solution for an oblique nonlinear problem suggested by the study of the Backus problem on the determination of the external gravitational potential

Pointwise andL2 bounds in some nonstandard problems for the heat equation

We consider some initial-boundary value problems for the linear and nonlinear heat eq where the gradient of the solution is prescribed on the boundary. Assuming that a solution we obtain bounds for

The spheroidal fixed–free two-boundary-value problem for geoid determination (the spheroidal Bruns' transform)

Abstract. The target of the spheroidal Gauss–Listing geoid determination is presented as a solution of the spheroidal fixed–free two-boundary value problem based on a spheroidal Bruns' transformation

A duality theorem for an overdetermined eigenvalue problem

We prove an equivalent integral formulation of an eigenvalue problem for the Laplacian operator, with overdetermined boundary conditions. A special case of this problem, that with homogeneous Neumann



Some Applications of the Maximum Principle in the Problem of Torsional Creep

In this paper we derive a maximum principle for the nonlinear differential equation of torsional creep and use this principle to compute isoperimetric bounds for the maximum stress and the torsional

A Fully Nonlinear Boundary value Problem for the Laplace Equation in Dimension Two

We study the regularity up to the boundary of solutions to the boundary value problem:[math001] in D, ∣?u∣= g on &pardD, where D is the unit disc. This problem finds its application in the study of

Solution of Dirichlet's problem for the equation Δu =−1 in a convex region

Let u be a solution of the following boundary-value problem: u¦Γ = 0, where Γ is a closed convex curve and Δu = −1 in the region D bounded by Γ. Then u has only one local maximum, and all its level

Extension of two theorems of Payne to some nonlinear Dirichlet problems

In this paper we prove that under certain convexity and symmetry assumptions on a domain in the plane any positive solutionu, of Δu+λf(u)=0, in,D,u=0 on ∂D has only one interior critical point. This

The problem of dirichlet for quasilinear elliptic differential equations with many independent variables

  • J. Serrin
  • Mathematics
    Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
  • 1969
This paper is concerned with the existence of solutions of the Dirichlet problem for quasilinear elliptic partial differential equations of second order, the conclusions being in the form of

A symmetry problem in potential theory

The proof of this result is given in Section 1 ; in Section 3 we give various generalizations to elliptic differential equations other than (1). Before turning to the detailed arguments it will be of