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Isoperimetric inequalities in the parabolic obstacle problems

Díaz Díaz, Jesús Ildefonso and Mossino, J. (1992) Isoperimetric inequalities in the parabolic obstacle problems. Comptes Rendus de l'Académie des Sciences. Série I. Mathématique , 71 (3). pp. 233-266. ISSN 0764-4442

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Abstract

We are concerned with the parabolic obstacle problem ut+Au+cu≥f,u≥ψ, (ut+Au+cu−f)(u−ψ)=0inQ=(0,T)×Ω, u=ψ on Σ=(0,T)×∂Ω, u|t=0=u0 in Ω, A being a linear elliptic second-order operator in divergence form or a nonlinear `pseudo-Laplacian'. We give an isoperimetric inequality for the concentration of u−ψ around its maximum. Various consequences are given. In particular, it is proved that u−ψ vanishes after a finite time, under a suitable assumption on ψt+Aψ+cψ−f. Other applications are also given.
"These results are deduced from the study of the particular case ψ=0. In this case, we prove that, among all linear second-order elliptic operators A having ellipticity constant 1, all equimeasurable domains Ω, all equimeasurable functions f and u0, the choice giving the `most concentrated' solution around its maximum is: A=−Δ, Ω is a ball Ω˜, f and u0 are radially symmetric and decreasing along the radii of Ω˜.
"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∞ in the sense of the theory of accretive operators.


Item Type:Article
Uncontrolled Keywords:parabolic obstacle problem; pseudo-Laplacian; isoperimetric inequality; linear second order elliptic operators; pointwise comparison; accretive operators theory
Subjects:Sciences > Mathematics > Differential geometry
ID Code:16387
References:

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Deposited On:17 Sep 2012 09:07
Last Modified:07 Feb 2014 09:28

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