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Isoperimetric inequalities in parabolic equations. (English) Zbl 0652.35053
The authors consider the parabolic problems: \[ (1)\quad \partial u/\partial t-\sum^{N}_{i,j=1}(\partial /\partial x_ j)a_{ij}(t,x)\partial u/\partial x_ i+cu=f\quad in\quad (0,T)\times \Omega \] \[ u=0\quad on\quad (0,T)\times \partial \Omega;\quad u(0,.)=u_ 0\quad for\quad t=0 \] where \(\Omega\) is a bounded and regular domain in \(R^ N\) (N\(\geq 1)\), \(a_{j}\) satisfy the uniform ellipticity condition \(\sum^{N}_{i,j=1}a_{ij}(t,x)\xi_ i\xi_ j\geq | \xi |^ 2\) \(\forall \xi \in R^ N\) and c, \(u_ 0\) and f are nonnegative functions and \[ (2)\quad \partial U/\partial t-\Delta U=\underset \tilde{} f\quad in\quad (0,T)\times {\tilde \Omega};\quad U=0\quad on\quad (0,T)\times \partial {\tilde \Omega};\quad U(0,)=\underset \tilde{} u_ 0 \] where \({\tilde \Omega}\) is the ball of \(R^ N\), centered at the origin which has the same measure as \(\Omega\) and \(\underset \tilde{} u_ 0\) (resp. \(\underset \tilde{} f(t,.))\) is the rearrangement of \(u_ 0\) (resp. \(f(t,.)\)) in \({\tilde \Omega}\) which decreases along the radii. It is known [see C. Bandle, Isoperimetric inequalities and applications” (Pitman 1980; Zbl 0436.35063)] that every strong solution u of problems (1-2) satisfies \[ \forall t\in [0,T],\quad \forall r\in [1,+\infty],\quad \| u(t,.)\|_{L^ r(\Omega)}\leq \infty (t,.)\|_{L^ r({\tilde \Omega})}. \] In this paper, the authors give a direct proof of the previous inequality, valid for every weak solution of problems (1-2). They prove also some other inequalities related to the previous one.
Reviewer: A.Canada

MSC:
35K20 Initial-boundary value problems for second-order parabolic equations
35B35 Stability in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
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