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Several regularity results for nonlocal elliptic equations

by Hui Ph Yu

Institution:	University of Texas Austin
Department:
Degree:
Year:	2017
Keywords:	Regularity theory for elliptic equations; Nonlocal operators; Nonlocal elliptic equations; Diffusion processes; Diffusion process modeling; Caffarelli; Silvestre; Viscosity solutions; Smooth solutions; Integrability estimates; Theory of flat solutio
Posted:	2/1/2018 12:00:00 AM
Record ID:	2152506
Full text PDF:	http://hdl.handle.net/2152/47271

Abstract

Nonlocal elliptic equations have long been used by physicists and engineers to model diffusion processes involving jumps. Apart from several works from a probabilistic view, there had not been much development concerning their mathematical properties until the fundamental works of Caffarelli and Silvestre. Here we establish several results concerning the regularity of viscosity solutions to nonlocal elliptic equations. In particular, we show the existence of smooth solutions to two class of nonlocal fully nonlinear elliptic equations, an integrability estimate for the fractional order Hessian of solutions to nonlocal equations, as well as a theory of at solutions.Advisors/Committee Members: Caffarelli, Luis A. (advisor), Figalli, Alessio (committee member), Jin, Tianling (committee member), Pavlovic, Natasa (committee member), Vasseur, Alexis F (committee member).