Nonphysical free boundaries in variational elliptic problems 

Prof. Eduardo Teixeira
University of Central Florida

Abstract: I will discuss regularity estimates at interior critical points of solutions to degenerate elliptic problems of divergence form. The problem is connected to the famous C^p′ regularity conjecture. Namely, that the conjectured sharp regularity for a function whose p-laplacian is bounded is attained by a naive looking, radially symmetric example, whose p-laplacian is actually constant, begs the question: is there any further regularity left for non-homogeneous models? In this talk I shall discuss this question through a new prism; the so called nonphysical free boundaries. I will show that if the source term is away from zero, then (any) solution fails to be in C^p′+ε for all ε>0. I will also show that improved estimates are eventually available at critical points for which the source term vanish at a prescribed rate (which may depend on u and/or $\nabla u$).