# American Institute of Mathematical Sciences

December  2015, 35(12): 6031-6068. doi: 10.3934/dcds.2015.35.6031

## Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations

Received  February 2014 Revised  August 2014 Published  May 2015

In this work we consider the problems $$\left\{\begin{array}{rcll} \mathcal{L \,} u&=&f &\hbox{ in } \Omega,\\ u&=&0 &\hbox{ in } \mathbb{R}^N\setminus\Omega, \end{array} \right.$$ and $$\left\{\begin{array}{rcll} u_t +\mathcal{L \,} u&=&f &\hbox{ in } Q_{T}\equiv\Omega\times (0, T),\\ u (x,t) &=&0 &\hbox{ in } \big(\mathbb{R}^N\setminus\Omega\big) \times (0, T),\\ u(x,0)&=&0 &\hbox{ in } \Omega, \end{array} \right.$$ where $\mathcal{L \,}$ is a nonlocal differential operator and $\Omega$ is a bounded domain in $\mathbb{R}^N$, with Lipschitz boundary.
The main goal of this work is to study existence, uniqueness and summability of the solution $u$ with respect to the summability of the datum $f$. In the process we establish an $L^p$-theory, for $p \geq 1$, associated to these problems and we prove some useful inequalities for the applications.
Citation: Tommaso Leonori, Ireneo Peral, Ana Primo, Fernando Soria. Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 6031-6068. doi: 10.3934/dcds.2015.35.6031
##### References:
 [1] B. Abdellaoui and I. Peral, Existence and nonexistence results for quasilinear elliptic equations involving the p-Laplacian with a critical potential, Ann. di Mat. Pura e Applicata, 182 (2003), 247-270. doi: 10.1007/s10231-002-0064-y. [2] R. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65, Academic Press, New York-London, 1975. [3] N. Alibaud, B. Andreianov and M. Bendahmane, Renormalized solutions of the fractional Laplace equation, C. R. Math. Acad. Sci. Paris, 348 (2010), 759-762. doi: 10.1016/j.crma.2010.05.006. [4] G. Alberti and G.Bellettini, A nonlocal anisotropic model for phase transitions. I. The optimal profile problem, Math. Ann., 310 (1998), 527-560. doi: 10.1007/s002080050159. [5] W. Allegretto and Y. X. Huang, A Picone's identity for the p-Laplacian and applications, Nonlinear Ana. T.M.P., 32 (1998), 819-830. doi: 10.1016/S0362-546X(97)00530-0. [6] D. Applebaum, Lévy Processes and Stochastic Calculus, $2^{nd}$ edition, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781. [7] D. G. Aronson and J. Serrin, Local behavior of solutions of quasilinear parabolic equations, Arch. Rational Mech. Anal., 25 (1967), 81-122. doi: 10.1007/BF00281291. [8] M. T. Barlow, R. F. Bass, Z.-Q. Chen and M. Kassmann, Non-local Dirichlet forms and symmetric jump processes, Trans. Amer. Math. Soc., 361 (2009), 1963-1999. doi: 10.1090/S0002-9947-08-04544-3. [9] B. Barrios, E. Colorado, R. Servadei and F. Soria, A critical fractional equation with concave-convex nonlinearities, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, in press, corrected proof, available online 2 May 2014. doi: 10.1016/j.anihpc.2014.04.003. [10] B. Barrios, M. Medina and I. Peral, Some remarks on the solvability of non-local elliptic problems with the Hardy potential, Commun. Contemp. Math., 16 (2014), 1350046, 29 pp. doi: 10.1142/S0219199713500466. [11] B. Barrios, I. Peral and S. Vita, Some remarks about the summability of nonlocal nonlinear problems, Advances in Nonlinear Analysis, Published online February 2015. doi: 10.1515/anona-2015-0012. [12] L. Boccardo, A. Dall'Aglio, T. Gallouët and L. Orsina, Existence and regularity results for some nonlinear parabolic equations, Adv. Math. Sci. Appl., 9 (1999), 1017-1031. [13] L. Boccardo and T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87 (1989), 149-169. doi: 10.1016/0022-1236(89)90005-0. [14] L. Boccardo, M. M. Porzio and A. Primo, Summability and existence results for nonlinear parabolic equations, Nonlinear Anal., 71 (2009), 978-990. doi: 10.1016/j.na.2008.11.066. [15] L. Boccardo and G. Croce, Esistenza e Regolarità di Soluzioni di Alcuni Problemi Ellitici, Quaderni dell'UMI. 51, Bologna, 2010. [16] L. Brasco and G. Franzina, Convexity properties of Dirichlet integrals and Picone-type inequalities, Kodai Math. J., 37 (2014), 769-799. doi: 10.2996/kmj/1414674621. [17] H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbb{R}^{N}$, Manuscripta Math., 74 (1992), 87-106. doi: 10.1007/BF02567660. [18] H. Brezis and P. Mironescu, Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces, J. Evol. Equ., 1 (2001), 387-404. doi: 10.1007/PL00001378. [19] H. Brezis and L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal., 10 (1986), 55-64. doi: 10.1016/0362-546X(86)90011-8. [20] X. Cabré and J. Sola-Morales, Layer solutions in a half-space for boundary reactions, Comm. Pure Appl. Math., 58 (2005), 1678-1732. doi: 10.1002/cpa.20093. [21] X. Cabré and J. M. Roquejoffre, The influence of fractional diffusion in Fisher-KPP equations, Comm. Math. Phys., 320 (2013), 679-722. doi: 10.1007/s00220-013-1682-5. [22] L. Caffarelli, Further regularity for the Signorini problem, Comm. Partial Differential Equations, 4 (1979), 1067-1075. doi: 10.1080/03605307908820119. [23] L. Caffarelli, J. M. Roquejoffre and Y. Sire, Variational problems in free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1151-1179. doi: 10.4171/JEMS/226. [24] L. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Annals of Mathematics. Second Series, 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903. [25] S. Campanato, Sistemi Ellittici in Forma Divergenza. Regolaritá All'interno, Quaderni Scuola Normale Superiore di Pisa, Pisa, 1980. [26] W. Craig and M. Groves, Hamiltonian long-wave approximations to the water-wave problem, Wave Motion, 19 (1994), 367-389. doi: 10.1016/0165-2125(94)90003-5. [27] W. Craig, U. Schanz and C. Sulem, The modulational regime of three-dimensional water waves and the Davey-Stewartson system, Ann. Inst. H. Poincare, Anal. Non Lineaire, 14 (1997), 615-667. doi: 10.1016/S0294-1449(97)80128-X. [28] R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, 5, Springer-Verlag, 1992. doi: 10.1007/978-3-642-58090-1. [29] A. Di Castro, T. Kuusi and G. Palatucci, Nonlocal Harnack inequalities, J. Funct. Anal., 267 (2014), 1807-1836. doi: 10.1016/j.jfa.2014.05.023. [30] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004. [31] S. Dipierro, G. Palatucci and E. Valdinoci, Dislocation dynamics in crystals: A macroscopic theory in a fractional Laplace setting, Commun. Math. Phys., 333 (2015), 1061-1105. doi: 10.1007/s00220-014-2118-6. [32] M. Felsinger and M. Kassmann, Local regularity for parabolic nonlocal operators, Comm. PDE, 38 (2013), 1539-1573. doi: 10.1080/03605302.2013.808211. [33] R. L. Frank, E. H. Lieb and R. Seiringer, Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators, J. Amer. Math. Soc., 21 (2008), 925-950. doi: 10.1090/S0894-0347-07-00582-6. [34] M. Fukushima, On an $L^p$-Estimate of Resolvents of Markov Processes, Publ. RIMS, Kyoto Univ., 13 (1977), 277-284. doi: 10.2977/prims/1195190108. [35] L. Hörmander, The Analysis of Linear Partial Differential Operators. III. Pseudo-differential Operators, Reprint of the 1994 edition, Classics in Mathematics, Springer, Berlin, 2007. [36] K. H. Karlsen, F. Petitta and S. Ulusoy, A duality approach to the fractional Laplacian with measure data, Publ. Mat., 55 (2011), 151-161. doi: 10.5565/PUBLMAT_55111_07. [37] M. Kassmann, A priori estimates for integro-differential operators with measurable kernels, Calc. Var., 34 (2009), 1-21. doi: 10.1007/s00526-008-0173-6. [38] M. Kassmann, Harnack inequality: An introduction, Bound. Value Probl., 2007, Art. ID 81415, 21 pp. [39] T.Kuusi, G.Mingione and Y. Sire, Nonlocal equations with measure data, Preprint available at cvgmt.sns.it. [40] E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032. [41] A. Majda and E. Tabak, A two-dimensional model for quasigeostrophic flow: Comparison with the two-dimensional Euler flow, Nonlinear Phenomena in Ocean Dynamics (Los Alamos, NM, 1995), Phys. D., 98 (1996), 515-522. doi: 10.1016/0167-2789(96)00114-5. [42] V. Maz'ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations, Second edition, Grundlehren der Mathematischen Wissenschaften, 342, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-15564-2. [43] 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. [44] A. N. Milgram, Supplement II in Partial Differential Equations, (eds. L. Bers, F. John and M. Schechter), Lectures in Applied Mathematics, Vol. III, Interscience New York, 1964. [45] G. Palatucci and A. Pisante, Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces, Calc. Var. Partial Differential Equations, 50 (2014), 799-829. doi: 10.1007/s00526-013-0656-y. [46] G. Palatucci, O. Savin and E. Valdinoci, Local and global minimizers for a variational energy involving a fractional norm, Annali di Matematica Pura ed Applicata, 192 (2013), 673-718. doi: 10.1007/s10231-011-0243-9. [47] M. Picone, Sui valori eccezionali di un parametro da cui dipende una equazione differenziale lineare ordinaria del secondo ordine, Ann. Scuola. Norm. Pisa., 11 (1910), p144. [48] X. Ros-Oton and J. Serra, The extremal solution for the fractional Laplacian, Calc. Var. Partial Differential Equations, 50 (2014), 723-750. doi: 10.1007/s00526-013-0653-1. [49] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003. [50] O. Savin and E. Valdinoci, Elliptic PDEs with fibered nonlinearities, J. Geom. Anal., 19 (2009), 420-432. doi: 10.1007/s12220-008-9064-5. [51] R. Servadei and E. Valdinoci, Mountain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898. doi: 10.1016/j.jmaa.2011.12.032. [52] A. Signorini, Questioni di elasticitá non linearizzata e semilinearizzata, Rendiconti di Matematica e delle sue applicazioni, 18 (1959), 95-139. [53] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Communications on Pure and Applied Mathematics, 60 (2007), 67-112. doi: 10.1002/cpa.20153. [54] 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. [55] Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864. doi: 10.1016/j.jfa.2009.01.020. [56] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189-258. doi: 10.5802/aif.204. [57] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J. 1970. [58] E. M. Stein and G. Weiss, Fractional integrals on n-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514. [59] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. [60] J. Stoker, Water Waves: The Mathematical Theory with Applications, Pure and Applied Mathematics, Vol. IV. Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1957. [61] M. E. Taylor, Tools for PDE. Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials, Mathematical Surveys and Monographs, 81, American Mathematical Society, Providence, RI, 2000. [62] J. Toland, The Peierls-Nabarro and Benjamin-Ono equations, J. Funct. Anal., 145 (1997), 136-150. doi: 10.1006/jfan.1996.3016. [63] E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl. $S\veceMA$, 49 (2009), 33-44. [64] M. I. Vishik, Mixed boundary problems, Dokl. Akad. Nauk SSSR, 97 (1954), 193-196.

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##### References:
 [1] B. Abdellaoui and I. Peral, Existence and nonexistence results for quasilinear elliptic equations involving the p-Laplacian with a critical potential, Ann. di Mat. Pura e Applicata, 182 (2003), 247-270. doi: 10.1007/s10231-002-0064-y. [2] R. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65, Academic Press, New York-London, 1975. [3] N. Alibaud, B. Andreianov and M. Bendahmane, Renormalized solutions of the fractional Laplace equation, C. R. Math. Acad. Sci. Paris, 348 (2010), 759-762. doi: 10.1016/j.crma.2010.05.006. [4] G. Alberti and G.Bellettini, A nonlocal anisotropic model for phase transitions. I. The optimal profile problem, Math. Ann., 310 (1998), 527-560. doi: 10.1007/s002080050159. [5] W. Allegretto and Y. X. Huang, A Picone's identity for the p-Laplacian and applications, Nonlinear Ana. T.M.P., 32 (1998), 819-830. doi: 10.1016/S0362-546X(97)00530-0. [6] D. Applebaum, Lévy Processes and Stochastic Calculus, $2^{nd}$ edition, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781. [7] D. G. Aronson and J. Serrin, Local behavior of solutions of quasilinear parabolic equations, Arch. Rational Mech. Anal., 25 (1967), 81-122. doi: 10.1007/BF00281291. [8] M. T. Barlow, R. F. Bass, Z.-Q. Chen and M. Kassmann, Non-local Dirichlet forms and symmetric jump processes, Trans. Amer. Math. Soc., 361 (2009), 1963-1999. doi: 10.1090/S0002-9947-08-04544-3. [9] B. Barrios, E. Colorado, R. Servadei and F. Soria, A critical fractional equation with concave-convex nonlinearities, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, in press, corrected proof, available online 2 May 2014. doi: 10.1016/j.anihpc.2014.04.003. [10] B. Barrios, M. Medina and I. Peral, Some remarks on the solvability of non-local elliptic problems with the Hardy potential, Commun. Contemp. Math., 16 (2014), 1350046, 29 pp. doi: 10.1142/S0219199713500466. [11] B. Barrios, I. Peral and S. Vita, Some remarks about the summability of nonlocal nonlinear problems, Advances in Nonlinear Analysis, Published online February 2015. doi: 10.1515/anona-2015-0012. [12] L. Boccardo, A. Dall'Aglio, T. Gallouët and L. Orsina, Existence and regularity results for some nonlinear parabolic equations, Adv. Math. Sci. Appl., 9 (1999), 1017-1031. [13] L. Boccardo and T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87 (1989), 149-169. doi: 10.1016/0022-1236(89)90005-0. [14] L. Boccardo, M. M. Porzio and A. Primo, Summability and existence results for nonlinear parabolic equations, Nonlinear Anal., 71 (2009), 978-990. doi: 10.1016/j.na.2008.11.066. [15] L. Boccardo and G. Croce, Esistenza e Regolarità di Soluzioni di Alcuni Problemi Ellitici, Quaderni dell'UMI. 51, Bologna, 2010. [16] L. Brasco and G. Franzina, Convexity properties of Dirichlet integrals and Picone-type inequalities, Kodai Math. J., 37 (2014), 769-799. doi: 10.2996/kmj/1414674621. [17] H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbb{R}^{N}$, Manuscripta Math., 74 (1992), 87-106. doi: 10.1007/BF02567660. [18] H. Brezis and P. Mironescu, Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces, J. Evol. Equ., 1 (2001), 387-404. doi: 10.1007/PL00001378. [19] H. Brezis and L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal., 10 (1986), 55-64. doi: 10.1016/0362-546X(86)90011-8. [20] X. Cabré and J. Sola-Morales, Layer solutions in a half-space for boundary reactions, Comm. Pure Appl. Math., 58 (2005), 1678-1732. doi: 10.1002/cpa.20093. [21] X. Cabré and J. M. Roquejoffre, The influence of fractional diffusion in Fisher-KPP equations, Comm. Math. Phys., 320 (2013), 679-722. doi: 10.1007/s00220-013-1682-5. [22] L. Caffarelli, Further regularity for the Signorini problem, Comm. Partial Differential Equations, 4 (1979), 1067-1075. doi: 10.1080/03605307908820119. [23] L. Caffarelli, J. M. Roquejoffre and Y. Sire, Variational problems in free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1151-1179. doi: 10.4171/JEMS/226. [24] L. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Annals of Mathematics. Second Series, 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903. [25] S. Campanato, Sistemi Ellittici in Forma Divergenza. Regolaritá All'interno, Quaderni Scuola Normale Superiore di Pisa, Pisa, 1980. [26] W. Craig and M. Groves, Hamiltonian long-wave approximations to the water-wave problem, Wave Motion, 19 (1994), 367-389. doi: 10.1016/0165-2125(94)90003-5. [27] W. Craig, U. Schanz and C. Sulem, The modulational regime of three-dimensional water waves and the Davey-Stewartson system, Ann. Inst. H. Poincare, Anal. Non Lineaire, 14 (1997), 615-667. doi: 10.1016/S0294-1449(97)80128-X. [28] R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, 5, Springer-Verlag, 1992. doi: 10.1007/978-3-642-58090-1. [29] A. Di Castro, T. Kuusi and G. Palatucci, Nonlocal Harnack inequalities, J. Funct. Anal., 267 (2014), 1807-1836. doi: 10.1016/j.jfa.2014.05.023. [30] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004. [31] S. Dipierro, G. Palatucci and E. Valdinoci, Dislocation dynamics in crystals: A macroscopic theory in a fractional Laplace setting, Commun. Math. Phys., 333 (2015), 1061-1105. doi: 10.1007/s00220-014-2118-6. [32] M. Felsinger and M. Kassmann, Local regularity for parabolic nonlocal operators, Comm. PDE, 38 (2013), 1539-1573. doi: 10.1080/03605302.2013.808211. [33] R. L. Frank, E. H. Lieb and R. Seiringer, Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators, J. Amer. Math. Soc., 21 (2008), 925-950. doi: 10.1090/S0894-0347-07-00582-6. [34] M. Fukushima, On an $L^p$-Estimate of Resolvents of Markov Processes, Publ. RIMS, Kyoto Univ., 13 (1977), 277-284. doi: 10.2977/prims/1195190108. [35] L. Hörmander, The Analysis of Linear Partial Differential Operators. III. Pseudo-differential Operators, Reprint of the 1994 edition, Classics in Mathematics, Springer, Berlin, 2007. [36] K. H. Karlsen, F. Petitta and S. Ulusoy, A duality approach to the fractional Laplacian with measure data, Publ. Mat., 55 (2011), 151-161. doi: 10.5565/PUBLMAT_55111_07. [37] M. Kassmann, A priori estimates for integro-differential operators with measurable kernels, Calc. Var., 34 (2009), 1-21. doi: 10.1007/s00526-008-0173-6. [38] M. Kassmann, Harnack inequality: An introduction, Bound. Value Probl., 2007, Art. ID 81415, 21 pp. [39] T.Kuusi, G.Mingione and Y. Sire, Nonlocal equations with measure data, Preprint available at cvgmt.sns.it. [40] E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032. [41] A. Majda and E. Tabak, A two-dimensional model for quasigeostrophic flow: Comparison with the two-dimensional Euler flow, Nonlinear Phenomena in Ocean Dynamics (Los Alamos, NM, 1995), Phys. D., 98 (1996), 515-522. doi: 10.1016/0167-2789(96)00114-5. [42] V. Maz'ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations, Second edition, Grundlehren der Mathematischen Wissenschaften, 342, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-15564-2. [43] 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. [44] A. N. Milgram, Supplement II in Partial Differential Equations, (eds. L. Bers, F. John and M. Schechter), Lectures in Applied Mathematics, Vol. III, Interscience New York, 1964. [45] G. Palatucci and A. Pisante, Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces, Calc. Var. Partial Differential Equations, 50 (2014), 799-829. doi: 10.1007/s00526-013-0656-y. [46] G. Palatucci, O. Savin and E. Valdinoci, Local and global minimizers for a variational energy involving a fractional norm, Annali di Matematica Pura ed Applicata, 192 (2013), 673-718. doi: 10.1007/s10231-011-0243-9. [47] M. Picone, Sui valori eccezionali di un parametro da cui dipende una equazione differenziale lineare ordinaria del secondo ordine, Ann. Scuola. Norm. Pisa., 11 (1910), p144. [48] X. Ros-Oton and J. Serra, The extremal solution for the fractional Laplacian, Calc. Var. Partial Differential Equations, 50 (2014), 723-750. doi: 10.1007/s00526-013-0653-1. [49] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003. [50] O. Savin and E. Valdinoci, Elliptic PDEs with fibered nonlinearities, J. Geom. Anal., 19 (2009), 420-432. doi: 10.1007/s12220-008-9064-5. [51] R. Servadei and E. Valdinoci, Mountain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898. doi: 10.1016/j.jmaa.2011.12.032. [52] A. Signorini, Questioni di elasticitá non linearizzata e semilinearizzata, Rendiconti di Matematica e delle sue applicazioni, 18 (1959), 95-139. [53] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Communications on Pure and Applied Mathematics, 60 (2007), 67-112. doi: 10.1002/cpa.20153. [54] 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. [55] Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864. doi: 10.1016/j.jfa.2009.01.020. [56] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189-258. doi: 10.5802/aif.204. [57] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J. 1970. [58] E. M. Stein and G. Weiss, Fractional integrals on n-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514. [59] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. [60] J. Stoker, Water Waves: The Mathematical Theory with Applications, Pure and Applied Mathematics, Vol. IV. Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1957. [61] M. E. Taylor, Tools for PDE. Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials, Mathematical Surveys and Monographs, 81, American Mathematical Society, Providence, RI, 2000. [62] J. Toland, The Peierls-Nabarro and Benjamin-Ono equations, J. Funct. Anal., 145 (1997), 136-150. doi: 10.1006/jfan.1996.3016. [63] E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl. $S\veceMA$, 49 (2009), 33-44. [64] M. I. Vishik, Mixed boundary problems, Dokl. Akad. Nauk SSSR, 97 (1954), 193-196.
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