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, 2015, 35 (12) : 6031-6068. doi: 10.3934/dcds.2015.35.6031
References:
[1]

Ann. di Mat. Pura e Applicata, 182 (2003), 247-270. doi: 10.1007/s10231-002-0064-y.  Google Scholar

[2]

Pure and Applied Mathematics, Vol. 65, Academic Press, New York-London, 1975.  Google Scholar

[3]

C. R. Math. Acad. Sci. Paris, 348 (2010), 759-762. doi: 10.1016/j.crma.2010.05.006.  Google Scholar

[4]

Math. Ann., 310 (1998), 527-560. doi: 10.1007/s002080050159.  Google Scholar

[5]

Nonlinear Ana. T.M.P., 32 (1998), 819-830. doi: 10.1016/S0362-546X(97)00530-0.  Google Scholar

[6]

$2^{nd}$ edition, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781.  Google Scholar

[7]

Arch. Rational Mech. Anal., 25 (1967), 81-122. doi: 10.1007/BF00281291.  Google Scholar

[8]

Trans. Amer. Math. Soc., 361 (2009), 1963-1999. doi: 10.1090/S0002-9947-08-04544-3.  Google Scholar

[9]

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.  Google Scholar

[10]

Commun. Contemp. Math., 16 (2014), 1350046, 29 pp. doi: 10.1142/S0219199713500466.  Google Scholar

[11]

Advances in Nonlinear Analysis, Published online February 2015. doi: 10.1515/anona-2015-0012.  Google Scholar

[12]

Adv. Math. Sci. Appl., 9 (1999), 1017-1031.  Google Scholar

[13]

J. Funct. Anal., 87 (1989), 149-169. doi: 10.1016/0022-1236(89)90005-0.  Google Scholar

[14]

Nonlinear Anal., 71 (2009), 978-990. doi: 10.1016/j.na.2008.11.066.  Google Scholar

[15]

Quaderni dell'UMI. 51, Bologna, 2010. Google Scholar

[16]

Kodai Math. J., 37 (2014), 769-799. doi: 10.2996/kmj/1414674621.  Google Scholar

[17]

Manuscripta Math., 74 (1992), 87-106. doi: 10.1007/BF02567660.  Google Scholar

[18]

J. Evol. Equ., 1 (2001), 387-404. doi: 10.1007/PL00001378.  Google Scholar

[19]

Nonlinear Anal., 10 (1986), 55-64. doi: 10.1016/0362-546X(86)90011-8.  Google Scholar

[20]

Comm. Pure Appl. Math., 58 (2005), 1678-1732. doi: 10.1002/cpa.20093.  Google Scholar

[21]

Comm. Math. Phys., 320 (2013), 679-722. doi: 10.1007/s00220-013-1682-5.  Google Scholar

[22]

Comm. Partial Differential Equations, 4 (1979), 1067-1075. doi: 10.1080/03605307908820119.  Google Scholar

[23]

J. Eur. Math. Soc., 12 (2010), 1151-1179. doi: 10.4171/JEMS/226.  Google Scholar

[24]

Annals of Mathematics. Second Series, 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903.  Google Scholar

[25]

Quaderni Scuola Normale Superiore di Pisa, Pisa, 1980.  Google Scholar

[26]

Wave Motion, 19 (1994), 367-389. doi: 10.1016/0165-2125(94)90003-5.  Google Scholar

[27]

Ann. Inst. H. Poincare, Anal. Non Lineaire, 14 (1997), 615-667. doi: 10.1016/S0294-1449(97)80128-X.  Google Scholar

[28]

5, Springer-Verlag, 1992. doi: 10.1007/978-3-642-58090-1.  Google Scholar

[29]

J. Funct. Anal., 267 (2014), 1807-1836. doi: 10.1016/j.jfa.2014.05.023.  Google Scholar

[30]

Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[31]

Commun. Math. Phys., 333 (2015), 1061-1105. doi: 10.1007/s00220-014-2118-6.  Google Scholar

[32]

Comm. PDE, 38 (2013), 1539-1573. doi: 10.1080/03605302.2013.808211.  Google Scholar

[33]

J. Amer. Math. Soc., 21 (2008), 925-950. doi: 10.1090/S0894-0347-07-00582-6.  Google Scholar

[34]

Publ. RIMS, Kyoto Univ., 13 (1977), 277-284. doi: 10.2977/prims/1195190108.  Google Scholar

[35]

Reprint of the 1994 edition, Classics in Mathematics, Springer, Berlin, 2007.  Google Scholar

[36]

Publ. Mat., 55 (2011), 151-161. doi: 10.5565/PUBLMAT_55111_07.  Google Scholar

[37]

Calc. Var., 34 (2009), 1-21. doi: 10.1007/s00526-008-0173-6.  Google Scholar

[38]

Bound. Value Probl., 2007, Art. ID 81415, 21 pp.  Google Scholar

[39]

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

[40]

Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032.  Google Scholar

[41]

Nonlinear Phenomena in Ocean Dynamics (Los Alamos, NM, 1995), Phys. D., 98 (1996), 515-522. doi: 10.1016/0167-2789(96)00114-5.  Google Scholar

[42]

Second edition, Grundlehren der Mathematischen Wissenschaften, 342, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-15564-2.  Google Scholar

[43]

Comm. Pure Appl. Math., 13 (1960), 457-468. doi: 10.1002/cpa.3160130308.  Google Scholar

[44]

(eds. L. Bers, F. John and M. Schechter), Lectures in Applied Mathematics, Vol. III, Interscience New York, 1964.  Google Scholar

[45]

Calc. Var. Partial Differential Equations, 50 (2014), 799-829. doi: 10.1007/s00526-013-0656-y.  Google Scholar

[46]

Annali di Matematica Pura ed Applicata, 192 (2013), 673-718. doi: 10.1007/s10231-011-0243-9.  Google Scholar

[47]

Ann. Scuola. Norm. Pisa., 11 (1910), p144.  Google Scholar

[48]

Calc. Var. Partial Differential Equations, 50 (2014), 723-750. doi: 10.1007/s00526-013-0653-1.  Google Scholar

[49]

J. Math. Pures Appl., 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003.  Google Scholar

[50]

J. Geom. Anal., 19 (2009), 420-432. doi: 10.1007/s12220-008-9064-5.  Google Scholar

[51]

J. Math. Anal. Appl., 389 (2012), 887-898. doi: 10.1016/j.jmaa.2011.12.032.  Google Scholar

[52]

Rendiconti di Matematica e delle sue applicazioni, 18 (1959), 95-139.  Google Scholar

[53]

Communications on Pure and Applied Mathematics, 60 (2007), 67-112. doi: 10.1002/cpa.20153.  Google Scholar

[54]

Indiana Univ. Math. J., 55 (2006), 1155-1174. doi: 10.1512/iumj.2006.55.2706.  Google Scholar

[55]

J. Funct. Anal., 256 (2009), 1842-1864. doi: 10.1016/j.jfa.2009.01.020.  Google Scholar

[56]

Ann. Inst. Fourier (Grenoble), 15 (1965), 189-258. doi: 10.5802/aif.204.  Google Scholar

[57]

Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J. 1970.  Google Scholar

[58]

J. Math. Mech., 7 (1958), 503-514.  Google Scholar

[59]

Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971.  Google Scholar

[60]

Pure and Applied Mathematics, Vol. IV. Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1957.  Google Scholar

[61]

Mathematical Surveys and Monographs, 81, American Mathematical Society, Providence, RI, 2000.  Google Scholar

[62]

J. Funct. Anal., 145 (1997), 136-150. doi: 10.1006/jfan.1996.3016.  Google Scholar

[63]

Bol. Soc. Esp. Mat. Apl. $S\veceMA$, 49 (2009), 33-44.  Google Scholar

[64]

Dokl. Akad. Nauk SSSR, 97 (1954), 193-196.  Google Scholar

show all references

References:
[1]

Ann. di Mat. Pura e Applicata, 182 (2003), 247-270. doi: 10.1007/s10231-002-0064-y.  Google Scholar

[2]

Pure and Applied Mathematics, Vol. 65, Academic Press, New York-London, 1975.  Google Scholar

[3]

C. R. Math. Acad. Sci. Paris, 348 (2010), 759-762. doi: 10.1016/j.crma.2010.05.006.  Google Scholar

[4]

Math. Ann., 310 (1998), 527-560. doi: 10.1007/s002080050159.  Google Scholar

[5]

Nonlinear Ana. T.M.P., 32 (1998), 819-830. doi: 10.1016/S0362-546X(97)00530-0.  Google Scholar

[6]

$2^{nd}$ edition, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781.  Google Scholar

[7]

Arch. Rational Mech. Anal., 25 (1967), 81-122. doi: 10.1007/BF00281291.  Google Scholar

[8]

Trans. Amer. Math. Soc., 361 (2009), 1963-1999. doi: 10.1090/S0002-9947-08-04544-3.  Google Scholar

[9]

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.  Google Scholar

[10]

Commun. Contemp. Math., 16 (2014), 1350046, 29 pp. doi: 10.1142/S0219199713500466.  Google Scholar

[11]

Advances in Nonlinear Analysis, Published online February 2015. doi: 10.1515/anona-2015-0012.  Google Scholar

[12]

Adv. Math. Sci. Appl., 9 (1999), 1017-1031.  Google Scholar

[13]

J. Funct. Anal., 87 (1989), 149-169. doi: 10.1016/0022-1236(89)90005-0.  Google Scholar

[14]

Nonlinear Anal., 71 (2009), 978-990. doi: 10.1016/j.na.2008.11.066.  Google Scholar

[15]

Quaderni dell'UMI. 51, Bologna, 2010. Google Scholar

[16]

Kodai Math. J., 37 (2014), 769-799. doi: 10.2996/kmj/1414674621.  Google Scholar

[17]

Manuscripta Math., 74 (1992), 87-106. doi: 10.1007/BF02567660.  Google Scholar

[18]

J. Evol. Equ., 1 (2001), 387-404. doi: 10.1007/PL00001378.  Google Scholar

[19]

Nonlinear Anal., 10 (1986), 55-64. doi: 10.1016/0362-546X(86)90011-8.  Google Scholar

[20]

Comm. Pure Appl. Math., 58 (2005), 1678-1732. doi: 10.1002/cpa.20093.  Google Scholar

[21]

Comm. Math. Phys., 320 (2013), 679-722. doi: 10.1007/s00220-013-1682-5.  Google Scholar

[22]

Comm. Partial Differential Equations, 4 (1979), 1067-1075. doi: 10.1080/03605307908820119.  Google Scholar

[23]

J. Eur. Math. Soc., 12 (2010), 1151-1179. doi: 10.4171/JEMS/226.  Google Scholar

[24]

Annals of Mathematics. Second Series, 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903.  Google Scholar

[25]

Quaderni Scuola Normale Superiore di Pisa, Pisa, 1980.  Google Scholar

[26]

Wave Motion, 19 (1994), 367-389. doi: 10.1016/0165-2125(94)90003-5.  Google Scholar

[27]

Ann. Inst. H. Poincare, Anal. Non Lineaire, 14 (1997), 615-667. doi: 10.1016/S0294-1449(97)80128-X.  Google Scholar

[28]

5, Springer-Verlag, 1992. doi: 10.1007/978-3-642-58090-1.  Google Scholar

[29]

J. Funct. Anal., 267 (2014), 1807-1836. doi: 10.1016/j.jfa.2014.05.023.  Google Scholar

[30]

Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[31]

Commun. Math. Phys., 333 (2015), 1061-1105. doi: 10.1007/s00220-014-2118-6.  Google Scholar

[32]

Comm. PDE, 38 (2013), 1539-1573. doi: 10.1080/03605302.2013.808211.  Google Scholar

[33]

J. Amer. Math. Soc., 21 (2008), 925-950. doi: 10.1090/S0894-0347-07-00582-6.  Google Scholar

[34]

Publ. RIMS, Kyoto Univ., 13 (1977), 277-284. doi: 10.2977/prims/1195190108.  Google Scholar

[35]

Reprint of the 1994 edition, Classics in Mathematics, Springer, Berlin, 2007.  Google Scholar

[36]

Publ. Mat., 55 (2011), 151-161. doi: 10.5565/PUBLMAT_55111_07.  Google Scholar

[37]

Calc. Var., 34 (2009), 1-21. doi: 10.1007/s00526-008-0173-6.  Google Scholar

[38]

Bound. Value Probl., 2007, Art. ID 81415, 21 pp.  Google Scholar

[39]

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

[40]

Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032.  Google Scholar

[41]

Nonlinear Phenomena in Ocean Dynamics (Los Alamos, NM, 1995), Phys. D., 98 (1996), 515-522. doi: 10.1016/0167-2789(96)00114-5.  Google Scholar

[42]

Second edition, Grundlehren der Mathematischen Wissenschaften, 342, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-15564-2.  Google Scholar

[43]

Comm. Pure Appl. Math., 13 (1960), 457-468. doi: 10.1002/cpa.3160130308.  Google Scholar

[44]

(eds. L. Bers, F. John and M. Schechter), Lectures in Applied Mathematics, Vol. III, Interscience New York, 1964.  Google Scholar

[45]

Calc. Var. Partial Differential Equations, 50 (2014), 799-829. doi: 10.1007/s00526-013-0656-y.  Google Scholar

[46]

Annali di Matematica Pura ed Applicata, 192 (2013), 673-718. doi: 10.1007/s10231-011-0243-9.  Google Scholar

[47]

Ann. Scuola. Norm. Pisa., 11 (1910), p144.  Google Scholar

[48]

Calc. Var. Partial Differential Equations, 50 (2014), 723-750. doi: 10.1007/s00526-013-0653-1.  Google Scholar

[49]

J. Math. Pures Appl., 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003.  Google Scholar

[50]

J. Geom. Anal., 19 (2009), 420-432. doi: 10.1007/s12220-008-9064-5.  Google Scholar

[51]

J. Math. Anal. Appl., 389 (2012), 887-898. doi: 10.1016/j.jmaa.2011.12.032.  Google Scholar

[52]

Rendiconti di Matematica e delle sue applicazioni, 18 (1959), 95-139.  Google Scholar

[53]

Communications on Pure and Applied Mathematics, 60 (2007), 67-112. doi: 10.1002/cpa.20153.  Google Scholar

[54]

Indiana Univ. Math. J., 55 (2006), 1155-1174. doi: 10.1512/iumj.2006.55.2706.  Google Scholar

[55]

J. Funct. Anal., 256 (2009), 1842-1864. doi: 10.1016/j.jfa.2009.01.020.  Google Scholar

[56]

Ann. Inst. Fourier (Grenoble), 15 (1965), 189-258. doi: 10.5802/aif.204.  Google Scholar

[57]

Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J. 1970.  Google Scholar

[58]

J. Math. Mech., 7 (1958), 503-514.  Google Scholar

[59]

Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971.  Google Scholar

[60]

Pure and Applied Mathematics, Vol. IV. Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1957.  Google Scholar

[61]

Mathematical Surveys and Monographs, 81, American Mathematical Society, Providence, RI, 2000.  Google Scholar

[62]

J. Funct. Anal., 145 (1997), 136-150. doi: 10.1006/jfan.1996.3016.  Google Scholar

[63]

Bol. Soc. Esp. Mat. Apl. $S\veceMA$, 49 (2009), 33-44.  Google Scholar

[64]

Dokl. Akad. Nauk SSSR, 97 (1954), 193-196.  Google Scholar

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