Thesis Defense - Johannes Hosle

Speaker: John Hosle

Title: 

MONOTONICITY FORMULAS IN POSITIVELY CURVED SETTINGS WITH

APPLICATIONS TO TWO-PHASE FREE BOUNDARY PROBLEMS

Abstract: 

Inspired by the monotonicity formula of Alt, Caffarelli, and Friedman, we consider a natural

variant of the ACF functional in various positively curved 2 dimensional settings where we integrate over

sublevel sets of Green’s function rather than disks. We prove sharp almost-monotonicity formulas for

our new functional Φ in the case of convex planar domains (both with pole on boundary and in interior)

and in the setting of 2 dimensional complete manifolds with Euclidean volume growth and nonnegative

Gaussian curvature. Some partial results in the planar log-concave weight setting are presented as well.

The tools include the Schwarz-Christoffel formula and Riesz decomposition using isothermal coordinates.

As a consequence, using quasiconformal techniques, we recover the Lipschitz bound of Gemmer, Moon,

and Raynor for solutions to the two-phase free boundary problem of Alt, Caffarelli, and Friedman up to a

fixed Neumann boundary. We also obtain a new Lipschitz bound in the manifold setting with constants

depending on only the total integral curvature, in contrast to work of Teixeira and Zhang, which gives

constants depending on pointwise bounds for the Riemann curvature tensor and its derivatives.