June  2018, 11(3): 547-559. doi: 10.3934/dcdss.2018030

Optimal elliptic regularity: A comparison between local and nonlocal equations

Dipartimento di Matematica e Informatica, Università degli Studi di Catania, Viale A. Doria 6 I-95125 Catania, Italy

Received  May 2017 Revised  August 2017 Published  October 2017

Fund Project: *The author was partially supported by Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (INdAM).

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 $δ>0$.

Citation: Sunra J. N. Mosconi. Optimal elliptic regularity: A comparison between local and nonlocal equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 547-559. doi: 10.3934/dcdss.2018030
References:
[1]

R. F. Bass and D. A. Levin, Harnack inequalities for jump processes, Potential Anal., 17 (2002), 375-388.  doi: 10.1023/A:1016378210944.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

S. A. Marano and S. Mosconi, Asymptotics for optimizers of the fractional Hardy-Sobolev inequality, preprint, arXiv: 1609.01869v3. Google Scholar

[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.  Google Scholar

[14]

J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math., 14 (1961), 577-591.  doi: 10.1002/cpa.3160140329.  Google Scholar

[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).   Google Scholar

[16]

J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math., 80 (1958), 931-954.  doi: 10.2307/2372841.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

R. F. Bass and D. A. Levin, Harnack inequalities for jump processes, Potential Anal., 17 (2002), 375-388.  doi: 10.1023/A:1016378210944.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

S. A. Marano and S. Mosconi, Asymptotics for optimizers of the fractional Hardy-Sobolev inequality, preprint, arXiv: 1609.01869v3. Google Scholar

[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.  Google Scholar

[14]

J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math., 14 (1961), 577-591.  doi: 10.1002/cpa.3160140329.  Google Scholar

[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).   Google Scholar

[16]

J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math., 80 (1958), 931-954.  doi: 10.2307/2372841.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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