On the curvature of some free boundaries in higher dimensions

TL;DR

This paper investigates the curvature properties of free boundaries in higher dimensions, extending known results from two dimensions, and introduces new methods to analyze the inner ball condition and related mappings.

Contribution

It establishes the equivalence of various formulations of the inner ball condition in higher dimensions and computes the Brouwer degree for a key boundary-related mapping.

Findings

Proved equivalence of different formulations of the inner ball condition in higher dimensions

Computed the Brouwer degree for a geometrically important boundary mapping

Provided a new proof of the inner ball condition in two dimensions

Abstract

It is known that any subharmonic quadrature domain in two dimensions satisfies a natural inner ball condition, in other words there is a specific upper bound on the curvature of the boundary. This result directly applies to free boundaries appearing in obstacle type problems and in Hele-Shaw flow. In the present paper we make partial progress on the corresponding question in higher dimensions. Specifically, we prove the equivalence between several different ways to formulate the inner ball condition, and we compute the Brouwer degree for a geometrically important mapping related to the Schwarz potential of the boundary. The latter gives in particular a new proof in the two dimensional case.