We show that the maximal Cheeger set of a Jordan domain Ω without necks is the union of all balls of radius r=h(Ω)^−1 contained in Ω. Here, h(Ω) denotes the Cheeger constant of Ω, that is, the infimum of the ratio of perimeter over area among subsets of Ω, and a Cheeger set is a set attaining the infimum. The radius r is shown to be the unique number such that the area of the inner parallel set Ωr is equal to π r^2. The proof of the main theorem requires the combination of several intermediate facts, some of which are of interest in their own right. Examples are given demonstrating the generality of the result as well as the sharpness of our assumptions. In particular, as an application of the main theorem, we illustrate how to effectively approximate the Cheeger constant of the Koch snowflake

Abstract. We prove a higher regularity result for the free boundary in the obstacle problem for the fractional Laplacian via a higher order boundary Harnack estimate.

We prove a sharp quantitative version of the p-Sobolev inequality for any 1

Quantitative isoperimetric inequalities are shown for anisotropic surface energies where the isoperimetric deficit controls both the Fraenkel asymmetry and a measure of the oscillation of the boundary with respect to the boundary of the corresponding Wulff shape.

We introduce and study certain variants of Gamow’s liquid drop model in which an anisotropic surface energy replaces the perimeter. After existence and nonexistence results are established, the shape of minimizers is analyzed. Under suitable regularity and ellipticity assumptions on the surface tension, Wulff shapes are minimizers in this problem if and only if the surface energy is isotropic. In sharp contrast, Wulff shapes are the unique minimizers for certain crystalline surface tensions. We also introduce and study several related liquid drop models with anisotropic repulsion for which the Wulff shape is the minimizer in the small mass regime.