G. A. Philippin

Some overdetermined boundary value problems with elliptical free boundaries

In this paper we study three different overdetermined boundary value problems in R2: a problem of torsion, a problem of electrostatic capacity, and a problem of polarization. In each case we prove that a solution exists if and only if the free boundary is an ellipse. The techniques we use rely on classical complex function theory, maximum principle, and some topological argument.

Blow-up in a class of non-linear parabolic problems with time-dependent coefficients under Robin type boundary conditions

The goal of this article is the determination of upper and lower bounds for the blow-up time t ☆ for a class of non-linear parabolic problems with time dependent coefficients under Robin type boundary conditions.

On a class of overdetermined eigenvalue problems

In this paper we present some new results of symmetry for inhomogeneous Dirichlet eigenvalue problems overdetermined by a condition involving the gradient of the first eigenfunction on the boundary. One specificity of the problem studied is the dependence of the equation and the boundary condition on the distance to the origin. The method of investigation is based on the use of continuous Steiner symmetrization together with some domain derivative tools. An application is given to the study of an overdetermined eigenvalue problem for a wedge-like membrane.

Decay bounds for solutions of second order parabolic problems and their derivatives IV

This article continues the derivation of explicit solution and gradient bounds for classes of second order nonlinear parabolic initial-boundary value problems. Earlier results were restricted to domains with boundaries of nonnegative average curvature. The purpose of the present article is to remove the curvature restriction and to extend the results to more general parabolic problems.

Some minimum principles for a class of elliptic boundary value problems

In this article, we establish a general inequality from which we derive some minimum principles valid for solutions of a class of elliptic boundary value problems.

On the Spatial Decay of Solutions to a Quasi-Linear Parabolic Initial-Boundary Value Problem and Their Derivatives

In this paper, we study the spatial decay of the solution of a quasi-linear heat equation in a long cylindrical region if the far end and the lateral surface are held at a zero temperature and a nonzero temperature is applied at the near end. Our result follows from the maximum principle applied to an auxiliary function

Explicit decay bounds in some quasilinear one-dimensional parabolic problems

\Phi

Some isoperimetric inequalities for capacity, polarization, and virtual mass

defined on the solution u and its derivatives.

Some overdetermined parabolic problems

In the first part of this paper we study a thermal diffusion process described by a semilinear parabolic problem and we introduce a new maximum principle in order to obtain explicit decay bounds for the temperature and its gradient. In the second part we find analogous bounds for the so-called ground water equation in a more general form.

Comparison results for solutions of reaction diffusion problems

This paper deals primarily with two classical problems of potential theory: the polarization problem of electrostatics and the virtual mass problem for the flow of a perfect fluid. The relationship between these problems and higher dimensional electrostatics problems (see e.g. [10, 15]) is exploited and, together with standard variational characterizations, used to derive isoperimetric inequalities for virtual mass and polarization.