Isoperimetric inequalities in the parabolic obstacle problems
We are concerned with the parabolic obstacle problem \[ \begin{cases} \left. \begin{aligned} &u_1 + \mathcal Au + cu \geq f, u \geq \psi \\ &(u_1 + \mathcal Au + cu -f)(u-\psi)=0 \end{aligned} \right\} \quad\text{in } Q=(oT)\times\Omega,\\ u = \psi \text{ on } \Sigma=(0,T)\times\hat\Omega,\\ u|_{t=0} = u_0 \text{ in } \Omega \end{cases} \] (\(\mathcal A\) is a linear second order elliptic operator in divergence form or a nonlinear ``pseudo-Laplacian'' and (\(u_ t=\frac{\partial u}{\partial t}\))). 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 \(\psi_ t+\mathcal A\psi +c\psi -f\). 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 \(\mathcal A\), having ellipticity constant 1, all equimeasurable domains \(\Omega\), all equimeasurable functions \(f\) and \(u_ 0\), the choice giving the ``most concentrated'' solution around its maximum is: \(\mathcal A=-\Delta\), \(\Omega\) is a ball \(\tilde \Omega\), f ad \(u_ 0\) are radially symmetric and decreasing along the radii of \(\tilde \Omega\). 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 \(L^\infty\) in the sense of the accretive operators theory.
scientific article; zbMATH DE number 4113370