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:
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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.  Google Scholar

[2]

R. Adams, Sobolev Spaces,, Pure and Applied Mathematics, (1975).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

D. Applebaum, Lévy Processes and Stochastic Calculus,, $2^{nd}$ edition, (2009).  doi: 10.1017/CBO9780511809781.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

L. Boccardo and G. Croce, Esistenza e Regolarità di Soluzioni di Alcuni Problemi Ellitici,, Quaderni dell'UMI. 51, (2010).   Google Scholar

[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.  Google Scholar

[17]

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[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[22]

L. Caffarelli, Further regularity for the Signorini problem,, Comm. Partial Differential Equations, 4 (1979), 1067.  doi: 10.1080/03605307908820119.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

M. Felsinger and M. Kassmann, Local regularity for parabolic nonlocal operators,, Comm. PDE, 38 (2013), 1539.  doi: 10.1080/03605302.2013.808211.  Google Scholar

[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.  Google Scholar

[34]

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

[35]

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

[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.  Google Scholar

[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.  Google Scholar

[38]

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

[39]

T.Kuusi, G.Mingione and Y. Sire, Nonlocal equations with measure data,, Preprint available at cvgmt.sns.it., ().   Google Scholar

[40]

E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, Ann. of Math., 118 (1983), 349.  doi: 10.2307/2007032.  Google Scholar

[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.  Google Scholar

[42]

V. Maz'ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations,, Second edition, (2011).  doi: 10.1007/978-3-642-15564-2.  Google Scholar

[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.  Google Scholar

[44]

A. N. Milgram, Supplement II in Partial Differential Equations,, (eds. L. Bers, (1964).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[50]

O. Savin and E. Valdinoci, Elliptic PDEs with fibered nonlinearities,, J. Geom. Anal., 19 (2009), 420.  doi: 10.1007/s12220-008-9064-5.  Google Scholar

[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.  Google Scholar

[52]

A. Signorini, Questioni di elasticitá non linearizzata e semilinearizzata,, Rendiconti di Matematica e delle sue applicazioni, 18 (1959), 95.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[55]

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