W σ,ϵ -estimates for nonlocal elliptic equations

Yu, Hui

doi:10.1016/j.anihpc.2016.09.003

W ^σ,ϵ-estimates for nonlocal elliptic equations

Yu, Hui

Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 5, pp. 1141-1153.

Résumé

We prove a $W^{σ, ϵ}$ -estimate for a class of nonlocal fully nonlinear elliptic equations by following Fanghua Lin's original approach [8] to the analogous problem for second order elliptic equations, by first proving a potential estimate, then combining this estimate with the ABP-type estimate by N. Guillen and R. Schwab to control the size of the superlevel sets of the σ-order derivatives of solutions.

MR Zbl

DOI : 10.1016/j.anihpc.2016.09.003

Mots clés : Nonlocal equations, Nonlinear elliptic equations, Potential estimate

@article{AIHPC_2017__34_5_1141_0,
     author = {Yu, Hui},
     title = {\protect\emph{W}         \protect\textsuperscript{            \protect\emph{\ensuremath{\sigma}},\protect\emph{\ensuremath{\epsilon}}         }-estimates for nonlocal elliptic equations},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1141--1153},
     publisher = {Elsevier},
     volume = {34},
     number = {5},
     year = {2017},
     doi = {10.1016/j.anihpc.2016.09.003},
     mrnumber = {3742518},
     zbl = {1378.35051},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2016.09.003/}
}

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Yu, Hui. W         ^σ,ϵ-estimates for nonlocal elliptic equations. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 5, pp. 1141-1153. doi : 10.1016/j.anihpc.2016.09.003. http://www.numdam.org/articles/10.1016/j.anihpc.2016.09.003/

Bibliographie
Cité par

[1] Armstrong, S.; Silvestre, L.; Smart, C. Partial regularity of solutions of fully nonlinear, uniformly elliptic equations, Commun. Pure Appl. Math., Volume 65 (2012), pp. 1169–1184 | DOI | MR | Zbl

[2] Caffarelli, L. Interior a priori estimates for solutions of fully nonlinear equations, Ann. Math., Volume 130 (1989), pp. 189–213 | DOI | MR | Zbl

[3] Caffarelli, L.; Cabré, X. Fully Nonlinear Elliptic Equations, America Mathematical Society Colloquium Publications, vol. 43, American Mathematical Society, Providence, RI, 1995 | DOI | MR | Zbl

[4] Caffarelli, L.; Silvestre, L. Regularity theory for fully nonlinear integro-differential equations, Commun. Pure Appl. Math., Volume 62 (2009), pp. 597–638 | DOI | MR | Zbl

[5] Evans, L. Some estimates for nondivergence structure, second order elliptic equations, Trans. Am. Math. Soc., Volume 287 (1985), pp. 701–712 | DOI | MR | Zbl

[6] E. Fabes, D. Stroock, The $L^{p}$ -integrability of Green's functions and fundamental solutions for elliptic and parabolic equations, IMA preprint 47, Univ. of Minnesota, 1983. | MR | Zbl

[7] Guillen, N.; Schwab, R. Aleksandrov–Bakelman–Pucci type estimates for integro-differential equations, Arch. Ration. Mech. Anal., Volume 206 (2012), pp. 111–157 | DOI | MR | Zbl

[8] Lin, F-H. Second derivative $L^{p}$ -estimate for elliptic equations of nondivergent type, Proc. Am. Math. Soc., Volume 96 (1986) no. 3, pp. 447–451 | MR | Zbl

[9] Yu, H. Smooth solutions to a class of nonlocal fully nonlinear elliptic equations, Indiana Math. J. (2016) (in press) | MR

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