Publication: Isoperimetric inequalities in the parabolic obstacle problems
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1992
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Mossino, J.
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Elsevier
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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.
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