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# Advances in Calculus of Variations

Managing Editor: Duzaar, Frank / Kinnunen, Juha

Editorial Board: Armstrong, Scott N. / Balogh, Zoltán / Cardiliaguet, Pierre / Dacorogna, Bernard / Dal Maso, Gianni / DiBenedetto, Emmanuele / Fonseca, Irene / Gianazza, Ugo / Ishii, Hitoshi / Kristensen, Jan / Manfredi, Juan / Martell, Jose Maria / Mingione, Giuseppe / Nystrom, Kaj / Riviére, Tristan / Schaetzle, Reiner / Shen, Zhongwei / Silvestre, Luis / Tonegawa, Yoshihiro / Touzi, Nizar / Wang, Guofang

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Volume 11, Issue 3

# On a new class of fractional partial differential equations II

Tien-Tsan Shieh
/ Daniel E. Spector
• Corresponding author
• Department of Applied Mathematics, National Chiao Tung University, Hsinchu,Taiwan; and National Center for Theoretical Sciences, National Taiwan University, No. 1 Sec. 4 Roosevelt Rd., Taipei,106, Taiwan
• Email
• Other articles by this author:
Published Online: 2017-04-19 | DOI: https://doi.org/10.1515/acv-2016-0056

## 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 ${L}^{1}$ Hardy inequality, obtain further regularity results for solutions of certain fractional PDE, demonstrate the existence of minimizers for integral functionals of the fractional gradient with non-linear dependence in the field, and also establish the existence of solutions to corresponding Euler–Lagrange equations obtained as conditions of minimality. In addition, we pose a number of open problems, the answers to which would fill in some gaps in the theory as well as to establish connections with more classical areas of study, including interpolation and the theory of Dirichlet forms.

MSC 2010: 26A33; 35R11; 49J45; 35JXX

Dedicated to Irene Fonseca, on the occasion of her 60th birthday, with esteem and affection

## 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. Google Scholar

• [2]

J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Grundlehren Math. Wiss. 223, Springer, Berlin, 1976. 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.

• [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. 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.

• [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.

• [7]

L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math. 62 (2009), no. 5, 597–638.

• [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.

• [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]

L. Caffarelli and J. L. Vazquez, Nonlinear porous medium flow with fractional potential pressure, Arch. Ration. Mech. Anal. 202 (2011), no. 2, 537–565.

• [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.

• [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. 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.

• [14]

A. Di Castro, T. Kuusi and G. Palatucci, Nonlocal Harnack inequalities, J. Funct. Anal. 267 (2014), 1807–1836.

• [15]

D. G. B. Edelen, A. E. Green and N. Laws, Nonlocal continuum mechanics, Arch. Ration. Mech. Anal. 43 (1971), no. 1, 36–44.

• [16]

D. G. B. Edelen and N. Laws, On the thermodynamics of systems with nonlocality, Arch. Ration. Mech. Anal. 43 (1971), 24–35.

• [17]

I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: ${L}^{p}$ Spaces, Springer Monogr. Math., Springer, New York, 2007. Google Scholar

• [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.

• [19]

M. Fukushima, On an ${L}^{p}$-estimate of resolvents of Markov processes, Publ. Res. Inst. Math. Sci. 13 (1977/78), no. 1, 277–284. Google Scholar

• [20]

M. Fukushima, Dirichlet Forms and Markov Processes, North-Holland Math. Libr. 23, North-Holland, Amsterdam, 1980. 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.

• [22]

J. Horváth, On some composition formulas, Proc. Amer. Math. Soc. 10 (1959), 433–437.

• [23]

T. Iwaniec and C. Sbordone, Riesz transforms and elliptic PDEs with VMO coefficients, J. Anal. Math. 74 (1998), 183–212.

• [24]

M. Kassmann, The theory of De Giorgi for non-local operators, C. R. Math. Acad. Sci. Paris 345 (2007), no. 11, 621–624.

• [25]

M. Kassmann, A priori estimates for integro-differential operators with measurable kernels, Calc. Var. Partial Differential Equations 34 (2009), no. 1, 1–21.

• [26]

D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Classics Appl. Math. 31, SIAM, Philadelphia, 2000. 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.

• [28]

T. Kuusi and G. Mingione, Universal potential estimates, J. Funct. Anal. 262 (2012), no. 10, 4205–4269.

• [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.

• [30]

T. Kuusi, G. Mingione and Y. Sire, Nonlocal self-improving properties, Anal. PDE 8 (2015), 57–114.

• [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.

• [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.

• [33]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science, Yverdon, 1993. Google Scholar

• [34]

A. Schikorra, Integro-differential harmonic maps into spheres, Comm. Partial Differential Equations 40 (2015), no. 3, 506–539.

• [35]

A. Schikorra, ${L}^{p}$-gradient harmonic maps into spheres and $\mathrm{SO}\left(N\right)$, Differential Integral Equations 28 (2015), no. 3–4, 383–408. Google Scholar

• [36]

A. Schikorra, ε-regularity for systems involving non-local, antisymmetric operators, Calc. Var. Partial Differential Equations 54 (2015), no. 4, 3531–3570.

• [37]

A. Schikorra, Nonlinear commutators for the fractional p-Laplacian and applications, Math. Ann. 366 (2016), no. 1–2, 695–720.

• [38]

A. Schikorra, T.-T. Shieh and D. Spector, ${L}^{p}$ theory for fractional gradient PDE with VMO coefficients, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 26 (2015), no. 4, 433–443. Google Scholar

• [39]

A. Schikorra, T.-T. Shieh and D. Spector, Regularity for a fractional p-Laplace equation, Commun. Contemp. Math. (2017), 10.1142/S0219199717500031.

• [40]

A. Schikorra, D. Spector and J. Van Schaftingen, An ${L}^{1}$-type estimate for Riesz potentials, preprint (2014), https://arxiv.org/abs/1411.2318.

• [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.

• [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. Google Scholar

• [43]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser. 30, Princeton University Press, Princeton, 1970. Google Scholar

• [44]

E. M. Stein and G. Weiss, Fractional integrals on n-dimensional Euclidean space, J. Math. Mech. 7 (1958), 503–514. Google Scholar

• [45]

E. M. Stein and G. Weiss, On the theory of harmonic functions of several variables. I. The theory of ${H}^{p}$-spaces, Acta Math. 103 (1960), 25–62. Google Scholar

• [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.

• [47]

R. S. Strichartz, Multipliers on fractional Sobolev spaces, J. Math. Mech. 16 (1967), 1031–1060. Google Scholar

• [48]

R. S. Strichartz, ${H}^{p}$ Sobolev spaces, Colloq. Math. 60/61 (1990), no. 1, 129–139. Google Scholar

Revised: 2017-03-14

Accepted: 2017-03-17

Published Online: 2017-04-19

Published in Print: 2018-07-01

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.

Citation Information: Advances in Calculus of Variations, Volume 11, Issue 3, Pages 289–307, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258,

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