Anisotropic Sobolev Capacity with Fractional Order

TL;DR

This paper introduces the anisotropic Sobolev capacity with fractional order, explores its properties, and applies it to characterize embeddings and geometric measures in anisotropic fractional Sobolev spaces.

Contribution

It develops the theory of anisotropic Sobolev capacity with fractional order and links it to geometric measures and embeddings in anisotropic fractional Sobolev spaces.

Findings

Established basic properties of anisotropic Sobolev capacity with fractional order.

Provided geometric characterizations for measures inducing Sobolev space embeddings.

Analyzed anisotropic fractional perimeter and asymptotic behavior of Minkowski inequality constants.

Abstract

In this paper, we introduce the anisotropic Sobolev capacity with fractional order and develop some basic properties for this new object. Applications to the theory of anisotropic fractional Sobolev spaces are provided. In particular, we give geometric characterizations for a nonnegative Radon measure $\mu$ that naturally induces an embedding of the anisotropic fractional Sobolev class $\dot{\Lambda}_{\alpha,K}^{1,1}$ into the $\mu$-based-Lebesgue-space $L^{n/\beta}_\mu$ with $0<\beta\le n$. Also, we investigate the anisotropic fractional $\alpha$-perimeter. Such a geometric quantity can be used to approximate the anisotropic Sobolev capacity with fractional order. Estimation on the constant in the related Minkowski inequality, which is asymptotically optimal as $\alpha\rightarrow 0^+$, will be provided.