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Optimal elliptic regularity: A comparison between local and nonlocal equations

*The author was partially supported by Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (INdAM).
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  • Given $L≥1$ , we discuss the problem of determining the highest $α=α(L)$ such that any solution to a homogeneous elliptic equation in divergence form with ellipticity ratio bounded by $L$ is in $C^α_{\rm loc}$ . This problem can be formulated both in the classical and non-local framework. In the classical case it is known that $α(L)≳ {\rm exp}(-CL^β)$ , for some $C, β≥q$ depending on the dimension $N≥q$ . We show that in the non-local case, $α(L)≳ L^{-1-δ}$ for all $δ>0$ .

    Mathematics Subject Classification: 6E35, 35B40, 49K22.


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  • [1] R. F. Bass and D. A. Levin, Harnack inequalities for jump processes, Potential Anal., 17 (2002), 375-388.  doi: 10.1023/A:1016378210944.
    [2] J. Bourgain, H. Brezis and P. Mironescu, Another look at Sobolev spaces, in Optimal Control and Partial Differential Equations (J. L. Menaldi, E. Rofman, A. Sulem Eds.), 439–455, IOS Press, Amsterdam, 2001.
    [3] L. BrascoE. Parini and M. Squassina, Stability of variational eigenvalues for the fractional $p$-Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1813-1845.  doi: 10.3934/dcds.2016.36.1813.
    [4] M. Camar-Eddine and P. Seppecher, Closure of the set of diffusion functionals with respect to Mosco-convergence, Math. Models Methods Appl. Sci., 12 (2002), 1153-1176.  doi: 10.1142/S0218202502002069.
    [5] E. De Giorgi, Sulla differernziabilitá e l'analiticitá delle estremali degli integrali multipli regolari, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 20 (1957), 438-441. 
    [6] E. Di Benedetto and N. S. Trudinger, Harnack inequalities for quasi-minima of variational integrals, Ann. Inst. Henri Poincaré Anal Non Linéaire, 1 (1984), 295-308.  doi: 10.1016/S0294-1449(16)30424-3.
    [7] A. Di CastroT. Kuusi and G. Palatucci, Nonlocal Harnack inequalities, J. Funct. Anal., 267 (2014), 1807-1836.  doi: 10.1016/j.jfa.2014.05.023.
    [8] E. B. Fabes and D. W. Stroock, A new proof of Moser's parabolic Harnack inequality using the old ideas of Nash, Arch. Rational Mech. Anal., 96 (1986), 327-338.  doi: 10.1007/BF00251802.
    [9] A. IannizzottoS. Mosconi and M. Squassina, Global Hölder regularity for the fractional $p$-Laplacian, Rev. Mat. Iberoam., 32 (2016), 1353-1392.  doi: 10.4171/RMI/921.
    [10] A. IannizzottoS. Mosconi and M. Squassina, A note on global regularity for the weak solutions of fractional $p-$Laplacian equations, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 27 (2016), 15-24.  doi: 10.4171/RLM/719.
    [11] M. Kassmann, A priori estimates for integro-differential operators with measurable kernels, Calc. Var. Partial Differential Equations, 34 (2009), 1-21.  doi: 10.1007/s00526-008-0173-6.
    [12] S. A. Marano and S. Mosconi, Asymptotics for optimizers of the fractional Hardy-Sobolev inequality, preprint, arXiv: 1609.01869v3.
    [13] J. Moser, A new proof of De Giorgi's theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math., 13 (1960), 457-468.  doi: 10.1002/cpa.3160130308.
    [14] J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math., 14 (1961), 577-591.  doi: 10.1002/cpa.3160140329.
    [15] F. Murat and L. Tartar, H-Convergence, in Topics in the Mathematical Modelling of Composite Materials (A. Cherkaev, R. Kohn Eds.), 21{43, Progr. Nonlinear Differential Equations Appl., 31, Birkh¨auser Boston, Boston, MA, (1997). 
    [16] J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math., 80 (1958), 931-954.  doi: 10.2307/2372841.
    [17] L. C. Piccinini and S. Spagnolo, On the Hölder continuity of solutions of second order elliptic equations in two variables, Ann. Scuola Norm. Sup. Pisa, 26 (1972), 391-402. 
    [18] L. Silvestre, Hölder estimates for solutions of integro-differential equations like the fractional Laplace, Indiana Univ. Math. J., 55 (2006), 1155-1174.  doi: 10.1512/iumj.2006.55.2706.
    [19] L. Silvestre and S. Snelson, An integro-differential equation without continuous solutions, Math. Res. Lett., 23 (2016), 1157-1166.  doi: 10.4310/MRL.2016.v23.n4.a9.
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