Geometric Measure Theory–Recent Applications

Abstract

Geometric Measure Theory (GMT) provides a framework to address questions in very different areas of mathematics, including calculus of variations, geometric analysis, potential theory, free boundary regularity, harmonic analysis, and theoretical computer science. Progress in different branches of GMT has led to the emergence of new challenges, making it a very vibrant area of research. In this note we will provide a historic background to some of the

Rectifiability; a survey

“…For the harmonic measure in the plane, see [GM05]. Toro's survey [Tor19] discusses relations between geometric measure theory and harmonic measure both in the plane and in higher dimensions.…”

Two Phase Free Boundary Problem for Poisson Kernels

“…Questions concerning the connections between the geometry of a domain and the regularity of its boundary with the potential theoretic properties of the domain, the behavior of singular integrals on the boundary, and the boundary regularity to solutions of elliptic PDEs have generated a flurry of activity in the area of nonsmooth analysis (see [Tor97] and [Tor19] for a brief recent history and references).…”