Regularity of fractional analogue of k-Hessian operators and a non-local one-phase free boundary problem
Access full-text files
Date
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
We study the regularity theory of fractional analogue of k-Hessian operators. We define the fractional k-Hessian operators as concave envelopes of linear fractional order operators. We have C¹,¹ regularity of viscosity solutions under the set-up of global solutions prescribing data at infinity and global barriers. Then we apply Evans-Krylov theorem to improve the regularity of fractional 2-Hessian operator to C [superscript 2s + alpha], and the key estimate is to prove the operator is strictly elliptic. We also study the minimizers of the energy [mathematical equation]. This non-local one-phase free boundary problem is an intermediate case of thin obstacle and fractional cavitation problem. We prove the homogeneity of the blow-up profiles and the regularity of free boundary under the flatness condition.