Workshop on Pdes 2022 - Teixeira

Nonphysical free boundaries in variational elliptic problems 

Prof. Eduardo Teixeira
University of Central Florida

Date: 15th February 13:30 pm - 14:15 pm (UTC -3) Link: PPGM/UFSCar Youtube channel

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 Cp′ 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 Cp′+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 $\nabla u$).