Convex symmetrization and applications
The paper deals with the following problem: How to replace a function \(u\in W^1_0(\mathbb{R}^n)\) with a ``symmetrized'' function \(u^*\) in order to have \[ \int(H(Du))^2dx\geq \int(H(Du^*))^2dx, \] where \(H:\mathbb{R}^n\to[0,\infty)\) is a convex function positively homogeneous of degree 1. It is well-known that, when \(H(\xi)=|\xi|\) then one can use Schwarz symmetrization to decrease the corresponding integral. For a general \(H(\xi)\), the authors consider the convex set \(K=\{x: H(x)\leq 1\}\), with \(H\) normalized so that the measure of \(K\) equals that of the unit ball in \(\mathbb{R}^n\). For a function \(u\in W^1_0(\mathbb{R}^n)\), its rearrangement with respect to \(H\) is defined as a function \(u^*\) whose level sets have the same measure of the corresponding level sets of \(u\) and are homothetic to \(K^0\), the polar set to \(K\). In this kind of ``symmetrization'', the convex set \(K^0\) plays the role of a ball in the Schwarz symmetrization. In the first part of the paper, a generalized perimeter \(P_H(E;\mathbb{R}^n)\) of a set \(E\) with respect to \(H\) is introduced. Then, the isoperimetric inequality \[ P_H(E;\mathbb{R}^n)\geq n|K^0|^{{1/n}}|E|^{1-{1/n}} \] is proved. In the second part, the Pólya-Szegö principle is obtained in the following form: for \(u\in W^{1,p}_0(\mathbb{R}^n)\), \(p\geq 1\), one has \[ \int H^p(Du)dx\geq \int H^p(Du^*)dx. \] In the last part, some applications are included. One example is the following. Let \(u\in W^1_0(\Omega)\) be a solution to the equation \(-(a_i(Du))_{x_i}= f\) in \(\Omega\), with \(a_i(\xi)\xi_i\geq (H(\xi))^2\), and let \(w\) satisfy \(-(H(Dw)H_{\xi_i}(Dw))_{x_i}= f^*\) in \(\Omega^*\), \(w=0\) on \(\partial\Omega^*\), where \(f^*\) is the rearrangement of \(f\) with respect to \(H\) and \(\Omega^*\) is the set homothetic to \(K^0\) and with the same measure as \(\Omega\). Then, the solution \(u(x)\), after symmetrization with respect to \(H\), satisfies \(u^*(x)\leq w(x)\).
scientific article; zbMATH DE number 1013374
scientific article