Abstract
In this manuscript we deal with existence/uniqueness and regularity issues of suitable weak solutions to nonlocal problems driven by fractional Laplace type operators. Different from previous researches, in our approach we consider gradient non-linearity sources with subcritical growth, as well as appropriated measures as sources and boundary datum. We provide an in-depth discussion on the notions of solutions involved together with existence/uniqueness results in different regimes and for different boundary value problems. Finally, this work extends previous ones by dealing with more general nonlocal operators, source terms and boundary data.
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References
Abdellaoui, B., Peral, I.: Towards a deterministic KPZ equation with fractional diffusion: the stationary problem. Nonlinearity 31(4), 1260–1298 (2018)
Abdellaoui, B., Peral, I.: Corrigendum: towards a deterministic KPZ equation with fractional diffusion: the stationary problem. Nonlinearity 33(3), C1–1 (2020)
Alibaud, N., Andreianov, B., Bendahmane, M.: Renormalized solutions of the fractional Laplace equation. C. R. Math. Acad. Sci. Paris 348(13–14), 759–762 (2010)
Applebaum, D.: Lévy processes—from probability to finance and quantum groups. Not. Am. Math. Soc. 51(11), 1336–1347 (2004)
Barrios, B., Figalli, A., Ros-Oton, X.: Global regularity for the free boundary in the obstacle problem for the fractional Laplacian. Am. J. Math. 140(2), 415–447 (2018)
Bénilan, Ph, Brezis, H.: Nonlinear problems related to the Thomas–Fermi equation. J. Evolut. Equ. 3, 673–770 (2003)
Bidaut-Véron, M.-F., Vivier, L.: An elliptic semilinear equation with source term involving boundary measures: the subcritical case. Rev. Mat. Iberoam. 16(3), 477–513 (2000)
Bjorland, C., Caffarelli, L., Figalli, A.: Non-local gradient dependent operators. Adv. Math. 230, 1859–1894 (2012)
Bjorland, C., Caffarelli, L., Figalli, A.: Nonlocal tug-of-war and the infinity fractional Laplacian. Commun. Pure Appl. Math. 65(3), 337–380 (2012)
Blumenthal, R.M., Getoor, R.K., Ray, D.B.: On the distribution of first hits for the symmetric stable processes. Trans. Am. Math. Soc. 99, 540–554 (1961)
Bogdan, K., Kulczycki, T., Nowak, A.: Gradient estimates for harmonic and \(q\)-harmonic functions of symmetric stable processes. Ill. J. Math. 46(2), 541–556 (2002)
Bogdan, K., Jakubowski, T.: Estimates of the Green function for the fractional Laplacian perturbed by gradient. Potential Anal. 36, 455–481 (2012)
Brezis, H.: Some variational problems of the Thomas-Fermi type. Variational inequalities and complementarity problems (Proc. Internat. School, Erice, 1978), pp. 53–73, Wiley, Chichester (1980)
Bucur, C.: Some observations on the Green function for the ball in the fractional Laplace framework. Commun. Pure Appl. Anal. 15(2), 657–699 (2016)
Bucur, C., Valdinocci, E.: Nonlocal Diffusion and Applications. Lecture Notes of the Unione Matematica Italiana, 20. Springer, [Cham]; Unione Matematica Italiana, Bologna, 2016. xii+155 pp. ISBN: 978-3-319-28738-6; 978-3-319-28739-3
Caffarelli, L.: Non local operators, drifts and games. Nonlinear PDEs. Abel Symposia 7, 37–52 (2012)
Caffarelli, L., Ros-Oton, X., Serra, J.: Obstacle problems for integro-differential operators: regularity of solutions and free boundaries. Invent. Math. 208(3), 1155–1211 (2017)
Caffarelli, L., Silvestre, L.: Regularity theory for fully-nonlinear integro-differential equations. Commun. Pure Appl. Math. 62, 597–638 (2009)
Caffarelli, L., Vasseur, A.: Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Ann. Math. 1903–1930 (2010)
Cantizano, N., Silva, A.: Three solutions for a nonlocal problem with critical growth. J. Math. Anal. Appl. 469(2), 841–851 (2019)
Chang, S.-Y.A., González, M.del M.: Fractional Laplacian in conformal geometry. Adv. Math. 226(2), 1410–1432 (2011)
Chen, H., Alhomedan, S., Hajaiej, H., Markowich, P.: Complete study of the existence and uniqueness of solutions for semilinear elliptic equations involving measures concentrated on boundary. Complex Var. Elliptic Equ. 62(12), 1687–1729 (2017)
Chen, H., Felmer, P., Véron, L.: Elliptic equations involving general subcritical source nonlinearity and measures (2014). \({<}\text{hal-}01072227{>}\)
Chen, H., Véron, L.: Semilinear fractional elliptic equations involving measures. J. Differ. Equ. 257(5), 1457–1486 (2014)
Chen, H., Véron, L.: Semilinear fractional elliptic equations with gradient nonlinearity involving measures. J. Funct. Anal. 266(8), 5467–5492 (2014)
Chen, Z.-Q., Song, R.: Estimates on Green functions and Poisson kernels for symmetric stable processes. Math. Ann. 312(3), 465–501 (1998)
Cont, R., Tankov, P.: Financial modelling with jump processes. Chapman & Hall/CRC Financial Mathematics Series. Chapman & Hall/CRC, Boca Raton, FL, 2004. xvi+535 pp. ISBN: 1-5848-8413-4
da Silva, J.V., Rossi, J.D.: The limit as \(p \rightarrow \infty \) in free boundary problems with fractional \(p\)-Laplacians. Trans. Am. Math. Soc. 371(4), 2739–2769 (2019)
da Silva, J.V., Salort, A.M.: A limiting obstacle type problem for the inhomogeneous \(p\)-fractional Laplacian. Calc. Var. Partial Differ. Equ. 58(4), 1–29 (2019)
Dipierro, S., Figalli, A., Valdinoci, E.: Strongly non local dislocation dynamics in crystals. Commun. Partial Differ. Equ. 39(12), 2351–2387 (2014)
Dipierro, S., Palatucci, G., Valdinoci, E.: Dislocation dynamics in crystals: a macroscopic theory in a fractional Laplace setting. Commun. Math. Phys. 333(2), 1061–1105 (2015)
Dipierro, S., Ros-Oton, X., Valdinoci, E.: Nonlocal problems with Neumann boundary conditions. Rev. Mat. Iberoam. 33(2), 377–416 (2017)
Fernández Bonder, J., Saintier, N., Silva, A.: The concentration-compactness principle for fractional order Sobolev spaces in unbounded domains and applications to the generalized fractional Brezis–Nirenberg problem. NoDEA Nonlinear Differ. Equ. Appl. 25(6), 1–25 (2018)
Fernández Bonder, J., Silva, A., Spedaletti, J.F.: Uniqueness of minimal energy solution for a semilinear problem involving the fractional Laplacian. Proc. Am. Math. Soc. 147(n07), 2925–2936 (2019)
Franzina, G., Valdinoci, E.: Geometric Analysis of Fractional Phase Transition Interfaces. Geometric Properties for Parabolic and Elliptic PDE’s. Springer INdAM Series, vol. 2, pp. 117–130. Springer, Milan (2013)
Galé, J.E., Miana, P.J., Stinga, P.R.: Extension problem and fractional operators: semigroups and wave equations. J. Evolut. Equ. 13(2), 343–368 (2013)
Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 224. Springer-Verlag, Berlin, 1983. xiii+513 pp. ISBN: 3-540-13025-X
Gilboa, G., Osher, S.: Non-local operators with applications to image processing. Multiscale Model. Simul. 7, 1005–1028 (2008)
Gkikas, K.T., Nguyen, P.T.: Semilinear elliptic equations with Hardy potential and gradient nonlinearity. Rev. Mat. Iberoam
Gkikas, K.T., Nguyen, P.T.: Elliptic equations with Hardy potential and gradient-dependent nonlinearity. Adv. Nonlinear Stud. 20(2), 399–435 (2020)
Gmira, A., Véron, L.: Boundary singularities of solutions of some nonlinear elliptic equations. Duke Math. J. 64, 271–324 (1991)
González, M.del M., Qing, J.: Fractional conformal Laplacians and fractional Yamabe problems. Anal. PDE 6(7), 1535–1576 (2013)
Kardar, M., Parisi, G., Zhang, Y.C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889–892 (1986)
Karlsen, K.H., Petitta, F., Ulusoy, S.: A duality approach to the fractional Laplacian with measure data. Publ. Mat. 55(1), 151–161 (2011)
Kulczycki, T., Properties of Green function of symmetric stable processes. Probab. Math. Statist. Acta Univ. Wratislav. No. 2029, 17(2), 339–364 (1997)
Kuusi, T., Mingione, G., Sire, Y.: Nonlocal equations with measure data. Commun. Math. Phys. 337(3), 1317–1368 (2015)
Kuusi, T., Mingione, G., Sire, Y.: Regularity issues involving the fractional \(p\)-Laplacian. In: Recent Developments in Nonlocal Theory, pp. 303–334. De Gruyter, Berlin, (2018)
Laskin, N.: Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A 268(4), 298–305 (2000)
Marcus, M., Nguyen, P.-T.: Elliptic equations with nonlinear absorption depending on the solution and its gradient. Proc. Lond. Math. Soc. (3) 111(1), 205–239 (2015)
Marcus, M., Véron, L.: Existence and uniqueness results for large solutions of general elliptic equations. J. Evolut. Equ. 3, 637–652 (2004)
Marcus, M., Véron, L.: Nonlinear second order elliptic equations involving measures. De Gruyter Series in Nonlinear Analysis and Applications, 21. De Gruyter, Berlin, 2014. xiv+248 pp. ISBN: 978-3-11-030515-9; 978-3-11-030531-9
Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)
Metzler, R., Klafter, J.: The restaurant at the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A 37, 161–208 (2004)
Nguyen, P.-T., Véron, L.: Boundary singularities of solutions to elliptic viscous Hamilton–Jacobi equations. J. Funct. Anal. 263(6), 1487–1538 (2012)
Nguyen, P.-T., Véron, L.: Boundary singularities of solutions to semilinear fractional equations. Adv. Nonlinear Stud. 18, 237–267 (2018)
Ros-Oton, X., Serra, J.: The Dirichlet problem for the fractional Laplacian: regularity up to the boundary. J. Math. Pures Appl. (9) 101(3), 275–302 (2014)
Silvestre, L.: Hölder estimates for solutions of integral-differential equations like the fractional Laplacian. Indiana Univ. Math. J. 55(3), 1155–1174 (2006)
Silvestre, L.: Regularity of the obstacle problem for a fractional power of the Laplace operator. Commun. Pure Appl. Math. 60(1), 67–112 (2007)
Silvestre, L.: On the differentiability of the solution to the Hamilton–Jacobi equation with critical fractional diffusion. Adv. Math. 226(2), 2020–2039 (2011)
Stinga, P.R.: User’s Guide to the Fractional Laplacian and the Method of Semigroups. Handbook of Fractional Calculus with Applications, vol. 2, pp. 235–265. De Gruyter, Berlin (2019)
Véron, L.: Elliptic equations involving measures. In: Chipot, M., Quittner, P. (eds.) Handbook of Differential Equations: Stationary Partial Differential Equations, vol. I, pp. 593–712. North-Holland, Amsterdam (2004)
Acknowledgements
We would like to thanks the referees for their useful corrections and suggestions. J.V. da Silva was partially supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (PNPD/CAPES-UnB-Brazil) Grant No. 88887.357992/2019-00, CNPq-Brazil under Grant No. 310303/2019-2 and FONCyT - PICT-2018-03183. A.S. is supported by PICT 2017-0704, by Universidad Nacional de San Luis under grants PROIPRO 03-2418 and PROICO 03-1916. P. O. is supported by Proyecto Bienal B080 Tipo 1 (Res. 4142/2019-R).
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This article is dedicated to the memory of Prof. Ireneo Peral.
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da Silva, J.V., Ochoa, P. & Silva, A. Fractional elliptic problems with nonlinear gradient sources and measures. Rev Mat Complut 35, 485–514 (2022). https://doi.org/10.1007/s13163-021-00391-1
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DOI: https://doi.org/10.1007/s13163-021-00391-1