Anisotropic symmetrization, convex bodies, and isoperimetric inequalities
Gabriele Bianchi, Andrea Cianchi, Paolo Gronchi
TL;DR
The paper addresses a fully anisotropic variant of the Pólya–Szegő principle for functionals of the form $\int_{\mathbb{R}^n} \Phi(\nabla u) \,dx$, where $\Phi$ is an $n$-dimensional Young function and symmetry is taken with respect to a convex body $K$.It develops a direct, non-approximation proof based on anisotropic isoperimetric inequalities and Brunn–Minkowski theory, establishing the universal inequality $\int_{\mathbb{R}^n} \Phi_{\bullet K\bullet}(\nabla u^{K}) \,dx \le \int_{\mathbb{R}^n} \Phi(\nabla u) \,dx$ for $u \in V^{1,\Phi}_{\mathrm{d}}(\mathbb{R}^n)$ and equality characterization.Equality analysis shows that when equality occurs, $u$ is quasi-convex and its level sets and gradients satisfy precise geometric and variational relations involving $\Phi_{\bullet}$ and the symmetrized functionals, yielding necessary and (under mild convexity assumptions) sufficient conditions.By unifying isotropic and anisotropic Pólya–Szegő principles within fully anisotropic Orlicz–Sobolev spaces and providing a geometric, extremal-aware framework, the work offers new tools for anisotropic variational problems and sharp shape optimization results.
Abstract
This work is concerned with a Pólya-Szegö type inequality for anisotropic functionals of Sobolev functions. The relevant inequality entails a double-symmetrization involving both trial functions and functionals. A new approach that uncovers geometric aspects of the inequality is proposed. It relies upon anisotropic isoperimetric inequalities, fine properties of Sobolev functions, and results from the Brunn-Minkowski theory of convex bodies. Importantly, unlike previously available proofs, the one offered in this paper does not require approximation arguments and hence allows for a characterization of extremal functions.