Weighted Dirichlet-type inequalities for Steiner symmetrization (Q1283878)

The following type of inequality is proved: \[ \int_{\mathbb{R}^{N}}F(x,u^{\ast },|\nabla u^{\ast }|) dx\leq \int_{\mathbb{R}^{N}}F(x,u,|\nabla u|) dx, \tag{1} \] where \(u\in W_{+}^{1,p}(\mathbb{R}^{N})\), \(1\leq p<\infty \). The function \(u^{\ast }\) is the Steiner symmetrization of \(u\) and \(F=F(x,u,z)\) is assumed to be a monotone nondecreasing and convex function in the last variable \(z\). This inequality was already proved in the case of other types of symmetrization, as well as for \(p>1\) and especially for \(F(x,u,z)=|z|^{p}\). See for related results: \textit{A. Alvino, P.-L. Lions} and \textit{G. Trombetti} [Nonlinear Anal., Theory Methods Appl. 13, No. 2, 185-220 (1989; Zbl 0678.49003)] and \textit{B. Kawohl} [Lecture Notes in Mathematics, 1150. Berlin etc.: Springer Verlag (1985; Zbl 0593.35002)], cited in the paper. In order to show \((1)\), some standard arguments are used like the approximation of \(W^{1,p}(\mathbb{R}^{N})\) by a subclass of the piecewise linear functions, some well-known elementary inequalities for convex functions in \(\mathbb{R}\), the weak compactness principle for sequences in \(L^{1}(\mathbb{R}^{N})\) and the weakly lower semicontinuity theorems in \(W^{1,p}(\mathbb{R}^{N})\). By means of \((1)\) it can be derived that the Steiner symmetrization is a mapping from \( W_{+}^{1,1}(\mathbb{R}^{N})\) into itself.