Browsing by Subject "Fractional Laplacian"
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Item Optimal regularity and nondegeneracy for minimizers of an energy related to the fractional Laplacian(2011-08) Yang, Ray; Caffarelli, Luis A.; Arapostathis, Ari; Beckner, William; Tsai, Richard; Vasseur, AlexisWe study the optimal regularity and nondegeneracy of a free boundary problem related to the fractional Laplacian through the extension technique of Caffarelli and Silvestre. Specifically, we show that minimizers of the energy [mathematical equation] where [mathematical equations] with 0 < [gamma] < 1, with free behavior on the set {y=0}, are Holder continuous with exponent [Beta] = 2[sigma]/2-[gamma]. These minimizers exhibit a free boundary: along {y = 0}, they divide into a zero set {u = 0} and a positivity set where {u > 0}; we call the interface between these sets the free boundary. The regularity is optimal, due to the non-degeneracy property of the minimizers: in any ball of radius r centered at the free boundary, the minimizer grows (in the supremum sense) like r[Beta]. This work is related to, but addresses a different problem from, recent work of Caffarelli, Roquejoffre, and Sire.Item Regularity of fractional analogue of k-Hessian operators and a non-local one-phase free boundary problem(2019-05-08) Wu, Yijing, Ph. D.; Caffarelli, Luis A.; Figalli, Alessio; Patrizi, Stefania; Vasseur, Alexis F.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.