Abstract: For $0<\theta <1$, the fractional $\theta$-perimeter of a set $E$ is defined as the Besov $B_{1,1}^{\theta }$-energy of the characteristic function of $E$, and  when multiplied by the scaling factor $(1-\theta )$, the $\theta$-perimeter is known in both Euclidean and metric settings to recover the classical perimeter as $\theta \to 1^{-}$.  In this talk, we show that in the setting of a complete doubling metric measure space supporting a $1$-Poincare inequality, the following fractional Poincare inequality holds, featuring this scaling factor for each $0<\theta <1$:

$\frac{1}{\mu (B)}{\int }_B|u-u_B|d\mu \le C(1-\theta )\frac{\text{rad}(B{)}^{\theta }}{\mu (\tau B)}{\int }_{\tau B}{\int }_{\tau B}\frac{|u(x)-u(y)|}{d(x,y{)}^{\theta }\mu (B(x,d(x,y)))}d\mu (y)d\mu (x).$

While such inequalities hold in the Euclidean setting, we show that their validity implies the $1$-Poincare inequality in doubling metric measure spaces.  We will also discuss the relationship between this fractional Poincare inequality and various isoperimetric and boxing inequalities given in terms of the $\theta$-perimeter, as well as some applications related to nonlocal minimal surfaces in the metric setting.  This talk is based on joint work with Panu Lahti, Jiang Li, and Xiaodan Zhou.

 

Link Information: https://us02web.zoom.us/j/87303952902

 

Organizer: Daniel Spector (NTNU)