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

1. 

Departamento de Análisis Matemático, Universidad de Granada, Avenida Fuentenueva S/N, 18071 GRANADA, Spain

2. 

Department of Mathematics, Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049 Madrid, Spain, Spain, Spain

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 & Continuous Dynamical Systems - A, 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. doi: 10.1007/s10231-002-0064-y.

[2]

R. Adams, Sobolev Spaces,, Pure and Applied Mathematics, (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. 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. 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. doi: 10.1016/S0362-546X(97)00530-0.

[6]

D. Applebaum, Lévy Processes and Stochastic Calculus,, $2^{nd}$ edition, (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. 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. 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, (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). 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, (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.

[13]

L. Boccardo and T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data,, J. Funct. Anal., 87 (1989), 149. 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. 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, (2010).

[16]

L. Brasco and G. Franzina, Convexity properties of Dirichlet integrals and Picone-type inequalities,, Kodai Math. J., 37 (2014), 769. doi: 10.2996/kmj/1414674621.

[17]

H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbbR^n$,, Manuscripta Math., 74 (1992), 87. doi: 10.1007/BF02567660.

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H. Brezis and P. Mironescu, Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces,, J. Evol. Equ., 1 (2001), 387. doi: 10.1007/PL00001378.

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H. Brezis and L. Oswald, Remarks on sublinear elliptic equations,, Nonlinear Anal., 10 (1986), 55. doi: 10.1016/0362-546X(86)90011-8.

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X. Cabré and J. Sola-Morales, Layer solutions in a half-space for boundary reactions,, Comm. Pure Appl. Math., 58 (2005), 1678. doi: 10.1002/cpa.20093.

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X. Cabré and J. M. Roquejoffre, The influence of fractional diffusion in Fisher-KPP equations,, Comm. Math. Phys., 320 (2013), 679. doi: 10.1007/s00220-013-1682-5.

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L. Caffarelli, Further regularity for the Signorini problem,, Comm. Partial Differential Equations, 4 (1979), 1067. 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. 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. doi: 10.4007/annals.2010.171.1903.

[25]

S. Campanato, Sistemi Ellittici in Forma Divergenza. Regolaritá All'interno,, Quaderni Scuola Normale Superiore di Pisa, (1980).

[26]

W. Craig and M. Groves, Hamiltonian long-wave approximations to the water-wave problem,, Wave Motion, 19 (1994), 367. 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, 14 (1997), 615. 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, 5 (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. 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. 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. doi: 10.1007/s00220-014-2118-6.

[32]

M. Felsinger and M. Kassmann, Local regularity for parabolic nonlocal operators,, Comm. PDE, 38 (2013), 1539. 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. doi: 10.1090/S0894-0347-07-00582-6.

[34]

M. Fukushima, On an $L^p$-Estimate of Resolvents of Markov Processes,, Publ. RIMS, 13 (1977), 277. doi: 10.2977/prims/1195190108.

[35]

L. Hörmander, The Analysis of Linear Partial Differential Operators. III. Pseudo-differential Operators,, Reprint of the 1994 edition, (1994).

[36]

K. H. Karlsen, F. Petitta and S. Ulusoy, A duality approach to the fractional Laplacian with measure data,, Publ. Mat., 55 (2011), 151. doi: 10.5565/PUBLMAT_55111_07.

[37]

M. Kassmann, A priori estimates for integro-differential operators with measurable kernels,, Calc. Var., 34 (2009), 1. doi: 10.1007/s00526-008-0173-6.

[38]

M. Kassmann, Harnack inequality: An introduction,, Bound. Value Probl., (2007).

[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. 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, 98 (1996), 515. doi: 10.1016/0167-2789(96)00114-5.

[42]

V. Maz'ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations,, Second edition, (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. doi: 10.1002/cpa.3160130308.

[44]

A. N. Milgram, Supplement II in Partial Differential Equations,, (eds. L. Bers, (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. 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. 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).

[48]

X. Ros-Oton and J. Serra, The extremal solution for the fractional Laplacian,, Calc. Var. Partial Differential Equations, 50 (2014), 723. 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. 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. 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. 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.

[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. 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. 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. 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. doi: 10.5802/aif.204.

[57]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton Mathematical Series, (1970).

[58]

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

[59]

E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces,, Princeton Mathematical Series, (1971).

[60]

J. Stoker, Water Waves: The Mathematical Theory with Applications,, Pure and Applied Mathematics, (1957).

[61]

M. E. Taylor, Tools for PDE. Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials,, Mathematical Surveys and Monographs, (2000).

[62]

J. Toland, The Peierls-Nabarro and Benjamin-Ono equations,, J. Funct. Anal., 145 (1997), 136. 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.

[64]

M. I. Vishik, Mixed boundary problems,, Dokl. Akad. Nauk SSSR, 97 (1954), 193.

show all references

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. doi: 10.1007/s10231-002-0064-y.

[2]

R. Adams, Sobolev Spaces,, Pure and Applied Mathematics, (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. 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. 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. doi: 10.1016/S0362-546X(97)00530-0.

[6]

D. Applebaum, Lévy Processes and Stochastic Calculus,, $2^{nd}$ edition, (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. 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. 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, (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). 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, (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.

[13]

L. Boccardo and T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data,, J. Funct. Anal., 87 (1989), 149. 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. 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, (2010).

[16]

L. Brasco and G. Franzina, Convexity properties of Dirichlet integrals and Picone-type inequalities,, Kodai Math. J., 37 (2014), 769. doi: 10.2996/kmj/1414674621.

[17]

H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbbR^n$,, Manuscripta Math., 74 (1992), 87. 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. doi: 10.1007/PL00001378.

[19]

H. Brezis and L. Oswald, Remarks on sublinear elliptic equations,, Nonlinear Anal., 10 (1986), 55. 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. 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. doi: 10.1007/s00220-013-1682-5.

[22]

L. Caffarelli, Further regularity for the Signorini problem,, Comm. Partial Differential Equations, 4 (1979), 1067. 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. 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. doi: 10.4007/annals.2010.171.1903.

[25]

S. Campanato, Sistemi Ellittici in Forma Divergenza. Regolaritá All'interno,, Quaderni Scuola Normale Superiore di Pisa, (1980).

[26]

W. Craig and M. Groves, Hamiltonian long-wave approximations to the water-wave problem,, Wave Motion, 19 (1994), 367. 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, 14 (1997), 615. 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, 5 (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. 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. 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. doi: 10.1007/s00220-014-2118-6.

[32]

M. Felsinger and M. Kassmann, Local regularity for parabolic nonlocal operators,, Comm. PDE, 38 (2013), 1539. 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. doi: 10.1090/S0894-0347-07-00582-6.

[34]

M. Fukushima, On an $L^p$-Estimate of Resolvents of Markov Processes,, Publ. RIMS, 13 (1977), 277. doi: 10.2977/prims/1195190108.

[35]

L. Hörmander, The Analysis of Linear Partial Differential Operators. III. Pseudo-differential Operators,, Reprint of the 1994 edition, (1994).

[36]

K. H. Karlsen, F. Petitta and S. Ulusoy, A duality approach to the fractional Laplacian with measure data,, Publ. Mat., 55 (2011), 151. doi: 10.5565/PUBLMAT_55111_07.

[37]

M. Kassmann, A priori estimates for integro-differential operators with measurable kernels,, Calc. Var., 34 (2009), 1. doi: 10.1007/s00526-008-0173-6.

[38]

M. Kassmann, Harnack inequality: An introduction,, Bound. Value Probl., (2007).

[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. 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, 98 (1996), 515. doi: 10.1016/0167-2789(96)00114-5.

[42]

V. Maz'ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations,, Second edition, (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. doi: 10.1002/cpa.3160130308.

[44]

A. N. Milgram, Supplement II in Partial Differential Equations,, (eds. L. Bers, (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. 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. 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).

[48]

X. Ros-Oton and J. Serra, The extremal solution for the fractional Laplacian,, Calc. Var. Partial Differential Equations, 50 (2014), 723. 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. 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. 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. 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.

[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. 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. 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. 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. doi: 10.5802/aif.204.

[57]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton Mathematical Series, (1970).

[58]

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

[59]

E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces,, Princeton Mathematical Series, (1971).

[60]

J. Stoker, Water Waves: The Mathematical Theory with Applications,, Pure and Applied Mathematics, (1957).

[61]

M. E. Taylor, Tools for PDE. Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials,, Mathematical Surveys and Monographs, (2000).

[62]

J. Toland, The Peierls-Nabarro and Benjamin-Ono equations,, J. Funct. Anal., 145 (1997), 136. 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.

[64]

M. I. Vishik, Mixed boundary problems,, Dokl. Akad. Nauk SSSR, 97 (1954), 193.

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