On the semi-additivity of the $1/2$-symmetric caloric capacity

TL;DR

This paper investigates a variant of the $1/2$-caloric capacity called the $1/2$-symmetric caloric capacity, establishing its semi-additivity and explicitly computing it for rectangles to reveal anisotropic properties.

Contribution

It introduces and analyzes the semi-additivity of the $1/2$-symmetric caloric capacity and provides explicit calculations for rectangles, highlighting its anisotropic nature.

Findings

Proved semi-additivity of the $1/2$-symmetric caloric capacity.

Explicitly computed the capacity for rectangles.

Revealed anisotropic behavior of the capacity.

Abstract

In this paper we study properties of a variant of the $1/2$-caloric capacity, called $1/2$-symmetric caloric capacity. The latter is associated simultaneously with the $1/2$-fractional heat equation and its conjugate. We establish its semi-additivity in $\mathbb{R}^{n+1}$ and, moreover, we compute explicitly the $1/2$-symmetric caloric capacity of rectangles, which illustrates its anisotropic behavior.