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Isoperimetric-inequalities in the parabolic obstacle problems

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Díaz Díaz, Jesús Ildefonso and Mossino, J. (1992) Isoperimetric-inequalities in the parabolic obstacle problems. Journal de Mathématiques Pures et Appliquées, 71 (3). pp. 233-266. ISSN 0021-7824

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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.


Item Type:Article
Uncontrolled Keywords:Stefan problem; rearrangement; parabolic obstacle problems; isoperimetric inequalities; comparison by rearrangement; accretive operators; extinction in finite time
Subjects:Sciences > Mathematics > Differential geometry
Sciences > Mathematics > Differential equations
ID Code:16260
Deposited On:10 Sep 2012 09:08
Last Modified:12 Dec 2018 15:08

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