American Institute of Mathematical Sciences

Advanced
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

Tommaso Leonori 1, , Ireneo Peral 2, , Ana Primo 2, and  Fernando Soria 2,

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.
Keywords: summability of the solutions., elliptic equations, Nonlocal operators, parabolic equations.
Mathematics Subject Classification: 45K05, 47G20, 35R09, 35D30, 35D3.
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

[1]

Nikolaos Roidos. Expanding solutions of quasilinear parabolic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021026
[2]

Arunima Bhattacharya, Micah Warren. $ C^{2, \alpha} $ estimates for solutions to almost Linear elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021024
[3]

Lidan Wang, Lihe Wang, Chunqin Zhou. Classification of positive solutions for fully nonlinear elliptic equations in unbounded cylinders. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1241-1261. doi: 10.3934/cpaa.2021019
[4]

Flank D. M. Bezerra, Jacson Simsen, Mariza Stefanello Simsen. Convergence of quasilinear parabolic equations to semilinear equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3823-3834. doi: 10.3934/dcdsb.2020258
[5]

Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199
[6]

Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327
[7]

Anderson L. A. de Araujo, Marcelo Montenegro. Existence of solution and asymptotic behavior for a class of parabolic equations. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1213-1227. doi: 10.3934/cpaa.2021017
[8]

Claudianor O. Alves, César T. Ledesma. Multiplicity of solutions for a class of fractional elliptic problems with critical exponential growth and nonlocal Neumann condition. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021058
[9]

Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617
[10]

Hongjie Dong, Xinghong Pan. On conormal derivative problem for parabolic equations with Dini mean oscillation coefficients. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021049
[11]

Zhang Chao, Minghua Yang. BMO type space associated with Neumann operator and application to a class of parabolic equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021104
[12]

Minh-Phuong Tran, Thanh-Nhan Nguyen. Pointwise gradient bounds for a class of very singular quasilinear elliptic equations. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021043
[13]

Xin-Guang Yang, Rong-Nian Wang, Xingjie Yan, Alain Miranville. Dynamics of the 2D Navier-Stokes equations with sublinear operators in Lipschitz-like domains. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3343-3366. doi: 10.3934/dcds.2020408
[14]

Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277
[15]

Livia Betz, Irwin Yousept. Optimal control of elliptic variational inequalities with bounded and unbounded operators. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021009
[16]

Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2725-3737. doi: 10.3934/dcds.2020383
[17]

Ankit Kumar, Kamal Jeet, Ramesh Kumar Vats. Controllability of Hilfer fractional integro-differential equations of Sobolev-type with a nonlocal condition in a Banach space. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021016
[18]

Marcel Braukhoff, Ansgar Jüngel. Entropy-dissipating finite-difference schemes for nonlinear fourth-order parabolic equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3335-3355. doi: 10.3934/dcdsb.2020234
[19]

Lianbing She, Nan Liu, Xin Li, Renhai Wang. Three types of weak pullback attractors for lattice pseudo-parabolic equations driven by locally Lipschitz noise. Electronic Research Archive, , () : -. doi: 10.3934/era.2021028
[20]

Bruno Premoselli. Einstein-Lichnerowicz type singular perturbations of critical nonlinear elliptic equations in dimension 3. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021069

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (196)
  • HTML views (0)
  • Cited by (51)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences