Isoperimetric-inequalities in the parabolic obstacle problems

Abstract

In this paper, we are concerned with the parabolic obstacle problem (u(t)=partial derivative u/partial derivative t [GRAPHICS] (A is a linear second order elliptic operator in divergence form or a nonlinear "pseudo-Laplacian"). We give an isoperimetric inequality for the concentration of u - psi around its maximum. Various consequences are given. In particular, it is proved that u - psi vanishes after a finite time, under a suitable assumption on psi(t) + A-psi + c- psi - f. Other applications are also given. These results are deduced from the study of the particular case psi=0. In this case, we prove that, among all linear second order elliptic operators A, having ellipticity constant 1, all equimeasurable domains OMEGA, all equimeasurable functions f and u0, the choice giving the "most concentrated" solution around its maximum is: A = -DELTA, OMEGA is a ball OMEGA, f and u0 are radially symmetric and decreasing along thc radii of OMEGA. A crucial point in our proof is a pointwise comparison result for an auxiliary one-dimensional unilateral problem. This is carried out by showing that this new problem is well-posed in L(infinity) in the sense of the accretive operators theory.