Fractional elliptic equations, Caccioppoli estimates and regularity
Annales de l'I.H.P. Analyse non linéaire, Volume 33 (2016) no. 3, pp. 767-807.

Let L=divx(A(x)x) be a uniformly elliptic operator in divergence form in a bounded domain Ω. We consider the fractional nonlocal equations

{Lsu=f,in Ω,u=0,on Ω,and{Lsu=f,in Ω,Au=0,on Ω.
Here Ls, 0<s<1, is the fractional power of L and Au is the conormal derivative of u with respect to the coefficients A(x). We reproduce Caccioppoli type estimates that allow us to develop the regularity theory. Indeed, we prove interior and boundary Schauder regularity estimates depending on the smoothness of the coefficients A(x), the right hand side f and the boundary of the domain. Moreover, we establish estimates for fundamental solutions in the spirit of the classical result by Littman–Stampacchia–Weinberger and we obtain nonlocal integro-differential formulas for Lsu(x). Essential tools in the analysis are the semigroup language approach and the extension problem.

DOI: 10.1016/j.anihpc.2015.01.004
Classification: 35R11, 35B65, 35K05, 35B45, 46E35
Keywords: Fractional Laplacian, Fractional divergence form elliptic operator, Schauder estimates, Fundamental solution, Semigroup language, Extension problem
     author = {Caffarelli, Luis A. and Stinga, Pablo Ra\'ul},
     title = {Fractional elliptic equations, {Caccioppoli} estimates and regularity},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {767--807},
     publisher = {Elsevier},
     volume = {33},
     number = {3},
     year = {2016},
     doi = {10.1016/j.anihpc.2015.01.004},
     zbl = {1381.35211},
     language = {en},
     url = {}
AU  - Caffarelli, Luis A.
AU  - Stinga, Pablo Raúl
TI  - Fractional elliptic equations, Caccioppoli estimates and regularity
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2016
SP  - 767
EP  - 807
VL  - 33
IS  - 3
PB  - Elsevier
UR  -
DO  - 10.1016/j.anihpc.2015.01.004
LA  - en
ID  - AIHPC_2016__33_3_767_0
ER  - 
%0 Journal Article
%A Caffarelli, Luis A.
%A Stinga, Pablo Raúl
%T Fractional elliptic equations, Caccioppoli estimates and regularity
%J Annales de l'I.H.P. Analyse non linéaire
%D 2016
%P 767-807
%V 33
%N 3
%I Elsevier
%R 10.1016/j.anihpc.2015.01.004
%G en
%F AIHPC_2016__33_3_767_0
Caffarelli, Luis A.; Stinga, Pablo Raúl. Fractional elliptic equations, Caccioppoli estimates and regularity. Annales de l'I.H.P. Analyse non linéaire, Volume 33 (2016) no. 3, pp. 767-807. doi : 10.1016/j.anihpc.2015.01.004.

[1] Allen, M.; Lindgren, E.; Petroshyan, A. The two-phase fractional obstacle problem, 2014 (27 pp) | arXiv

[2] Aronson, D.G. Bounds for the fundamental solution of a parabolic equation, Bull. Am. Math. Soc., Volume 73 (1967), pp. 890–896 | DOI | Zbl

[3] Auscher, P.; Tchamitchian, Ph. Evolution Equations and Their Applications in Physical and Life Sciences, Lecture Notes in Pure and Appl. Math., Volume vol. 215, Dekker, New York (2001), pp. 15–32 (Bad Herrenalb, 1998) | Zbl

[4] Cabré, X.; Tan, J. Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., Volume 224 (2010), pp. 2052–2093 | DOI | Zbl

[5] Caffarelli, L. Elliptic second order equations, Rend. Semin. Mat. Fis. Milano, Volume 58 (1988), pp. 253–284 | DOI | Zbl

[6] Caffarelli, L.; Salsa, S.; Silvestre, L. Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., Volume 171 (2008), pp. 425–461 | DOI | Zbl

[7] Caffarelli, L.; Silvestre, L. An extension problem related to the fractional Laplacian, Commun. Partial Differ. Equ., Volume 32 (2007), pp. 1245–1260 | DOI | Zbl

[8] Campanato, S. Proprietà di una famiglia di spazi funzionali, Ann. Sc. Norm. Super. Pisa (3), Volume 18 (1964), pp. 137–160 | Numdam | Zbl

[9] Capella, A.; Dávila, J.; Dupaigne, L.; Sire, Y. Regularity of radial extremal solutions for some non-local semilinear equations, Commun. Partial Differ. Equ., Volume 36 (2011), pp. 1353–1384 | DOI | Zbl

[10] Davies, E.B. Heat Kernels and Spectral Theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1989 | Zbl

[11] Fabes, E.B. Proceedings of the International Conference on Partial Differential Equations Dedicated to Luigi Amerio on His 70th Birthday, Rend. Semin. Mat. Fis. Milano, Volume vol. 52 (1982), pp. 11–21 (Milan/Como, 1982) | DOI | Zbl

[12] Fabes, E.B.; Jerison, D.S.; Kenig, C.E. The Wiener test for degenerate elliptic equations, Ann. Inst. Fourier (Grenoble), Volume 32 (1982), pp. 151–182 | DOI | Numdam | Zbl

[13] Fabes, E.B.; Kenig, C.E.; Serapioni, R.P. The local regularity of solutions of degenerate elliptic equations, Commun. Partial Differ. Equ., Volume 7 (1982), pp. 77–116 | DOI | Zbl

[14] Getoor, R.K. First passage times for symmetric stable processes in space, Trans. Am. Math. Soc., Volume 101 (1961), pp. 75–90 | DOI | Zbl

[15] Gilbarg, D.; Trudinger, N.S. Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001 (reprint of the 1998 edition) | Zbl

[16] Grubb, G. Fractional Laplacians on domains, a development of Hörmander's theory of μ-transmission pseudodifferential operators, Adv. Math., Volume 268 (2015), pp. 478–528 | DOI | Zbl

[17] Grubb, G. Regularity of spectral fractional Dirichlet and Neumann problems, 2014 (12 pp) | arXiv

[18] Han, Q.; Lin, F. Elliptic Partial Differential Equations, Courant Lecture Notes in Mathematics, vol. 1, Courant Institute of Mathematical Sciences, American Mathematical Society, New York, Providence, RI, 2011 | Zbl

[19] Lions, J.-L. Théorèmes de trace et d'interpolation (I), Ann. Sc. Norm. Super. Pisa (3), Volume 13 (1959), pp. 389–403 | Numdam | Zbl

[20] Lions, J.-L.; Magenes, E. Problèmes aux Limites Non Homogènes et Applications, vol. 1, Travaux et Recherches Mathématiques, vol. 17, Dunod, Paris, 1968 | Zbl

[21] Littman, W.; Stampacchia, G.; Weinberger, H.F. Regular points for elliptic equations with discontinuous coefficients, Ann. Sc. Norm. Super. Pisa (3), Volume 17 (1963), pp. 43–77 | Numdam | Zbl

[22] Nochetto, R.H.; Otárola, E.; Salgado, A.J. A PDE approach to fractional diffusion in general domains: a priori error analysis, Found. Comput. Math. (2014) (in press, 59 pp) | DOI

[23] Riahi, L. Estimates for Dirichlet heat kernels, intrinsic ultracontractivity and expected exit time on Lipschitz domains, Commun. Math. Anal., Volume 15 (2013), pp. 115–130 | Zbl

[24] Roncal, L.; Stinga, P.R. Fractional Laplacian on the torus, Commun. Contemp. Math. (2015) (in press)

[25] Ros-Otón, X.; Serra, J. The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl., Volume 101 (2014), pp. 275–302 | DOI | Zbl

[26] Saloff-Coste, L. The heat kernel and its estimates, Probabilistic Approach to Geometry, Adv. Stud. Pure Math., vol. 57, Math. Soc. Japan, Tokyo, 2010, pp. 405–436 | DOI | Zbl

[27] Seeley, R. Norms and domains of the complex powers ABz , Am. J. Math., Volume 93 (1971), pp. 299–309 | DOI | Zbl

[28] Seeley, R. Interpolation in Lp with boundary conditions, Stud. Math., Volume 44 (1972), pp. 47–60 | DOI | Zbl

[29] Silvestre, L. Regularity of the obstacle problem for a fractional power of the Laplace operator, Commun. Pure Appl. Math., Volume 60 (2007), pp. 67–112 | DOI | Zbl

[30] Song, R.; Vondraček, Z. Potential theory of subordinate killed Brownian motion in a domain, Probab. Theory Relat. Fields, Volume 125 (2003), pp. 578–592 | DOI | Zbl

[31] Stein, E.M. Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, 1970 | Zbl

[32] Stinga, P.R. Fractional powers of second order partial differential operators: extension problem and regularity theory, Universidad Autónoma de Madrid, Spain, 2010 (PhD thesis)

[33] Stinga, P.R.; Torrea, J.L. Extension problem and Harnack's inequality for some fractional operators, Commun. Partial Differ. Equ., Volume 35 (2010), pp. 2092–2122 | DOI | Zbl

[34] Stinga, P.R.; Volzone, B. Fractional semilinear Neumann problems arising from a fractional Keller–Segel model, Calc. Var. Partial Differ. Equ. (2014) (in press, 34 pp) | DOI

[35] Stinga, P.R.; Zhang, C. Harnack's inequality for fractional nonlocal equations, Discrete Contin. Dyn. Syst., Volume 33 (2013), pp. 3153–3170 | Zbl

[36] Turesson, B.O. Nonlinear Potential Theory and Weighted Sobolev Spaces, Lecture Notes in Mathematics, vol. 1736, Springer-Verlag, Berlin, 2000 | Zbl

[37] Zygmund, A. Smooth functions, Duke Math. J., Volume 12 (1945), pp. 47–76 | DOI | Zbl

Cited by Sources: