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.
Anisotropic symmetrization, convex bodies, and isoperimetric inequalities / Gabriele Bianchi; Andrea Cianchi; Paolo Gronchi. - In: ADVANCES IN MATHEMATICS. - ISSN 1090-2082. - STAMPA. - 462:(2025), pp. 110085.1-110085.36. [10.1016/j.aim.2024.110085]
Anisotropic symmetrization, convex bodies, and isoperimetric inequalities
Gabriele Bianchi;Andrea Cianchi;Paolo Gronchi
2025
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.File | Dimensione | Formato | |
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