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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.