Abstract
In this paper we continue to advance the theory regarding the Riesz fractional gradient in the calculus of variations and fractional partial differential equations begun in an earlier work of the same name. In particular, we here establish an
Dedicated to Irene Fonseca, on the occasion of her 60th birthday, with esteem and affection
Funding statement: The first author is partially supported by National Science Council of Taiwan under research grant NSC 101-2115-M-009-014-MY3. The second author is supported by the Taiwan Ministry of Science and Technology under research grants 103-2115-M-009-016-MY2 and 105-2115-M-009-004-MY2.
Acknowledgements
The first author would like to thank Professor Ming-Chih Lai for his support during a portion of this project. The second author would like to thank Eliot Fried for discussions regarding non-local continuum mechanics and also for making him aware of the work of Edelen and Laws, Edelen, Green and Laws. The authors would like to thank Armin Schikorra for many helpful discussions regarding the paper. Finally, the authors would like to thank the referees for their comments, which have improved the clarity of presentation in the paper.
References
[1] D. R. Adams and N. G. Meyers, Bessel potentials. Inclusion relations among classes of exceptional sets, Indiana Univ. Math. J. 22 (1972/73), 873–905. 10.1090/S0002-9904-1971-12821-5Search in Google Scholar
[2] J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Grundlehren Math. Wiss. 223, Springer, Berlin, 1976. 10.1007/978-3-642-66451-9Search in Google Scholar
[3] P. Biler, C. Imbert and G. Karch, The nonlocal porous medium equation: Barenblatt profiles and other weak solutions, Arch. Ration. Mech. Anal. 215 (2015), no. 2, 497–529. 10.1007/s00205-014-0786-1Search in Google Scholar
[4] S. Blatt, P. Reiter and A. Schikorra, Harmonic analysis meets critical knots. Critical points of the Möbius energy are smooth, Trans. Amer. Math. Soc. 368 (2016), no. 9, 6391–6438. 10.1090/tran/6603Search in Google Scholar
[5] L. Brasco and E. Lindgren, Higher Sobolev regularity for the fractional p-Laplace equation in the superquadratic case, Adv. Math. 304 (2017), 300–354. 10.1016/j.aim.2016.03.039Search in Google Scholar
[6] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), no. 7–9, 1245–1260. 10.1080/03605300600987306Search in Google Scholar
[7] L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math. 62 (2009), no. 5, 597–638. 10.1002/cpa.20274Search in Google Scholar
[8] L. Caffarelli and L. Silvestre, The Evans–Krylov theorem for nonlocal fully nonlinear equations, Ann. of Math. (2) 174 (2011), no. 2, 1163–1187. 10.4007/annals.2011.174.2.9Search in Google Scholar
[9] L. Caffarelli, F. Soria and J.-L. Vázquez, Regularity of solutions of the fractional porous medium flow, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 5, 1701–1746. 10.4171/JEMS/401Search in Google Scholar
[10] L. Caffarelli and J. L. Vazquez, Nonlinear porous medium flow with fractional potential pressure, Arch. Ration. Mech. Anal. 202 (2011), no. 2, 537–565. 10.1007/s00205-011-0420-4Search in Google Scholar
[11] L. A. Caffarelli and P. R. Stinga, Fractional elliptic equations, Caccioppoli estimates and regularity, Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), no. 3, 767–807. 10.1016/j.anihpc.2015.01.004Search in Google Scholar
[12] E. De Giorgi, Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Nat. (3) 3 (1957), 25–43. Search in Google Scholar
[13] A. Di Castro, T. Kuusi and G. Palatucci, Local behaviour of fractional p-minimizers, preprint (2015), https://arxiv.org/abs/1505.00361. Search in Google Scholar
[14] A. Di Castro, T. Kuusi and G. Palatucci, Nonlocal Harnack inequalities, J. Funct. Anal. 267 (2014), 1807–1836. 10.1016/j.jfa.2014.05.023Search in Google Scholar
[15] D. G. B. Edelen, A. E. Green and N. Laws, Nonlocal continuum mechanics, Arch. Ration. Mech. Anal. 43 (1971), no. 1, 36–44. 10.1007/BF00251544Search in Google Scholar
[16] D. G. B. Edelen and N. Laws, On the thermodynamics of systems with nonlocality, Arch. Ration. Mech. Anal. 43 (1971), 24–35. 10.1007/BF00251543Search in Google Scholar
[17]
I. Fonseca and G. Leoni,
Modern Methods in the Calculus of Variations:
[18] R. L. Frank and R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Anal. 255 (2008), no. 12, 3407–3430. 10.1016/j.jfa.2008.05.015Search in Google Scholar
[19]
M. Fukushima,
On an
[20] M. Fukushima, Dirichlet Forms and Markov Processes, North-Holland Math. Libr. 23, North-Holland, Amsterdam, 1980. Search in Google Scholar
[21] N. Guillen and R. W. Schwab, Aleksandrov–Bakelman–Pucci type estimates for integro-differential equations, Arch. Ration. Mech. Anal. 206 (2012), no. 1, 111–157. 10.1007/s00205-012-0529-0Search in Google Scholar
[22] J. Horváth, On some composition formulas, Proc. Amer. Math. Soc. 10 (1959), 433–437. 10.1090/S0002-9939-1959-0107788-4Search in Google Scholar
[23] T. Iwaniec and C. Sbordone, Riesz transforms and elliptic PDEs with VMO coefficients, J. Anal. Math. 74 (1998), 183–212. 10.1007/BF02819450Search in Google Scholar
[24] M. Kassmann, The theory of De Giorgi for non-local operators, C. R. Math. Acad. Sci. Paris 345 (2007), no. 11, 621–624. 10.1016/j.crma.2007.10.007Search in Google Scholar
[25] M. Kassmann, A priori estimates for integro-differential operators with measurable kernels, Calc. Var. Partial Differential Equations 34 (2009), no. 1, 1–21. 10.1007/s00526-008-0173-6Search in Google Scholar
[26] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Classics Appl. Math. 31, SIAM, Philadelphia, 2000. 10.1137/1.9780898719451Search in Google Scholar
[27] J. Korvenpää, T. Kuusi and G. Palatucci, The obstacle problem for nonlinear integro-differential operators, Calc. Var. Partial Differential Equations 55 (2016), no. 3, Article ID 63. 10.1007/s00526-016-0999-2Search in Google Scholar
[28] T. Kuusi and G. Mingione, Universal potential estimates, J. Funct. Anal. 262 (2012), no. 10, 4205–4269. 10.1016/j.jfa.2012.02.018Search in Google Scholar
[29] T. Kuusi, G. Mingione and Y. Sire, A fractional Gehring lemma, with applications to nonlocal equations, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Nat. IX. Ser. Rend. Lincei Mat. Appl. 25 (2014), 345–358. 10.4171/RLM/683Search in Google Scholar
[30] T. Kuusi, G. Mingione and Y. Sire, Nonlocal self-improving properties, Anal. PDE 8 (2015), 57–114. 10.2140/apde.2015.8.57Search in Google Scholar
[31] L. Martinazzi, A. Maalaoui and A. Schikorra, Blow-up behaviour of a fractional Adams–Moser–Trudinger type inequality in odd dimension, preprint (2015), https://arxiv.org/abs/1504.00254. Search in Google Scholar
[32] T. Mengesha and D. Spector, Localization of nonlocal gradients in various topologies, Calc. Var. Partial Differential Equations 52 (2015), no. 1–2, 253–279. 10.1007/s00526-014-0711-3Search in Google Scholar
[33] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science, Yverdon, 1993. Search in Google Scholar
[34] A. Schikorra, Integro-differential harmonic maps into spheres, Comm. Partial Differential Equations 40 (2015), no. 3, 506–539. 10.1080/03605302.2014.974059Search in Google Scholar
[35]
A. Schikorra,
[36] A. Schikorra, ε-regularity for systems involving non-local, antisymmetric operators, Calc. Var. Partial Differential Equations 54 (2015), no. 4, 3531–3570. 10.1007/s00526-015-0913-3Search in Google Scholar
[37] A. Schikorra, Nonlinear commutators for the fractional p-Laplacian and applications, Math. Ann. 366 (2016), no. 1–2, 695–720. 10.1007/s00208-015-1347-0Search in Google Scholar
[38]
A. Schikorra, T.-T. Shieh and D. Spector,
[39] A. Schikorra, T.-T. Shieh and D. Spector, Regularity for a fractional p-Laplace equation, Commun. Contemp. Math. (2017), 10.1142/S0219199717500031. 10.1142/S0219199717500031Search in Google Scholar
[40]
A. Schikorra, D. Spector and J. Van Schaftingen,
An
[41] T.-T. Shieh and D. Spector, On a new class of fractional partial differential equations, Adv. Calc. Var. 8 (2015), no. 4, 321–336. 10.1515/acv-2014-0009Search in Google Scholar
[42] S. L. Sobolev and S. M. Nikol’skiĭ, Embedding theorems, Proceedings of the Fourth All-Union Mathematical Congress. Vol. I (Leningrad 1961), Izdat. Akad. Nauk SSSR, Leningrad (1963), 227–242. Search in Google Scholar
[43] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser. 30, Princeton University Press, Princeton, 1970. 10.1515/9781400883882Search in Google Scholar
[44] E. M. Stein and G. Weiss, Fractional integrals on n-dimensional Euclidean space, J. Math. Mech. 7 (1958), 503–514. 10.1512/iumj.1958.7.57030Search in Google Scholar
[45]
E. M. Stein and G. Weiss,
On the theory of harmonic functions of several variables. I. The theory of
[46] P. R. Stinga and J. L. Torrea, Extension problem and Harnack’s inequality for some fractional operators, Comm. Partial Differential Equations 35 (2010), no. 11, 2092–2122. 10.1080/03605301003735680Search in Google Scholar
[47] R. S. Strichartz, Multipliers on fractional Sobolev spaces, J. Math. Mech. 16 (1967), 1031–1060. Search in Google Scholar
[48]
R. S. Strichartz,
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