Anisotropic Improved Leray-Trudinger Inequality

TL;DR

This paper extends the Leray-Trudinger inequality to anisotropic Sobolev spaces defined by a convex Finsler norm, providing a generalized exponential integrability result and establishing the optimal constant for radial functions.

Contribution

It introduces an anisotropic version of the Leray-Trudinger inequality using Finsler norms, generalizing classical results to a broader geometric setting.

Findings

Established anisotropic Leray-Trudinger inequality with Finsler norms.

Derived the optimal constant for anisotropically radial functions.

Generalized classical exponential integrability inequalities to anisotropic Sobolev spaces.

Abstract

We establish a Leray- Trudinger Type inequality in the anisotropic setting induced by a strongly convex Finsler norm F. The result generalizes classical exponential integrability inequalities for Sobolev functions to the framework of anisotropic Sobolev spaces $W^{1,n}_0(\Omega)$, where the standard Euclidean norm is replaced by F and associated polar norm $F^o$. Moreover, in the class of anisotropically radial functions, we obtain the optimal constant in the spirit of Moser's sharp inequality.