Regularity of fractional analogue of k-Hessian operators and a non-local one-phase free boundary problem

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2019-05-08

Authors

Wu, Yijing, Ph. D.

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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.

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