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September  2012, 11(5): 1897-1910. doi: 10.3934/cpaa.2012.11.1897

Local maximum principle for $L^p$-viscosity solutions of fully nonlinear elliptic PDEs with unbounded coefficients

Shigeaki Koike 1, and  Andrzej Świech 2,

1. 

Department of Mathematics, Saitama University, 255 Shimo-Okubo, Urawa, Saitama 338-8570
2. 

School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA 30332-0160

Received  March 2011 Revised  June 2011 Published  March 2012

We establish local maximum principle for $L^p$-viscosity solutions of fully nonlinear elliptic partial differential equations with unbounded ingredients.
Keywords: viscosity solutions, Fully nonlinear elliptic PDE, local maximum principle..
Mathematics Subject Classification: Primary: 45J15, 35J60, 35B5.
Citation: Shigeaki Koike, Andrzej Świech. Local maximum principle for $L^p$-viscosity solutions of fully nonlinear elliptic PDEs with unbounded coefficients. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1897-1910. doi: 10.3934/cpaa.2012.11.1897
References:
[1]

Abstr. Appl. Anal., (2008), Art. ID 178534, 19 pp.  Google Scholar

[2]

Comm. Pure Appl. Math., 48 (1995), 539-570. doi: 10.1002/cpa.3160480504.  Google Scholar

[3]

Ann. Math., 130 (1989), 189-213. doi: 10.2307/1971480.  Google Scholar

[4]

American Mathematical Society, Providence, 1995.  Google Scholar

[5]

Comm. Pure Appl. Math., 49 (1996), 365-397. doi: 10.1002/(SICI)1097-0312(199604)49:4<365::AID-CPA3>3.0.CO;2-A.  Google Scholar

[6]

Bull. Amer. Math. Soc., 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[7]

Comm. Partial Differential Equations, 25 (2000), 1997-2053. doi: 10.1080/03605300008821576.  Google Scholar

[8]

Indiana Univ. Math. J., 42 (1993), 413-423. doi: 10.1512/iumj.1993.42.42019.  Google Scholar

[9]

Duke Math. J., 51 (1984), 997-1016. doi: 10.1215/S0012-7094-84-05145-7.  Google Scholar

[10]

Ph.D. Thesis, UCSB, 1996. Google Scholar

[11]

Comm. Partial Differential Equations, 23 (1998), 967-983. doi: 10.1080/03605309808821375.  Google Scholar

[12]

2nd ed., Springer-Verlag, New York, 1983.  Google Scholar

[13]

J. Differential Equations, 250 (2011), 1553-1574. doi: 10.1016/j.jde.2010.07.005.  Google Scholar

[14]

Math. Ann., 339 (2007), 461-484. doi: 10.1007/s00208-007-0125-z.  Google Scholar

[15]

J. Math. Soc. Japan, 61 (2009), 723-755. doi: 10.2969/jmsj/06130723.  Google Scholar

[16]

(Russian) Dokl. Akad. Nauk SSSR, 245 (1979), 18-20.  Google Scholar

[17]

(Russian) Izv. Akad. Nauk SSSR Ser. Mat., 44 (1980), 161-175, 239.  Google Scholar

[18]

(Russian) in "Boundary Value Problems of Mathematical Physics and Related Questions in the Theory of Functions,'' 12, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI), 96 (1980), 272-287, 312.  Google Scholar

[19]

Arch. Ration. Mech. Anal., 195 (2010), 579-607 doi: 10.1007/s00205-009-0218-9.  Google Scholar

[20]

Invent. Math., 61 (1980), 67-79. doi: 10.1007/BF01389895.  Google Scholar

[21]

Rev. Mat. Iberoamericana, 4 (1988), 453-468. doi: 10.4171/RMI/80.  Google Scholar

[22]

Comm. Pure Appl. Math., 45 (1992), 27-76. doi: 10.1002/cpa.3160450103.  Google Scholar

[23]

Z. Anal. Anwend., 28 (2009), 129-164. doi: 10.4171/ZAA/1377.  Google Scholar

show all references

References:
[1]

Abstr. Appl. Anal., (2008), Art. ID 178534, 19 pp.  Google Scholar

[2]

Comm. Pure Appl. Math., 48 (1995), 539-570. doi: 10.1002/cpa.3160480504.  Google Scholar

[3]

Ann. Math., 130 (1989), 189-213. doi: 10.2307/1971480.  Google Scholar

[4]

American Mathematical Society, Providence, 1995.  Google Scholar

[5]

Comm. Pure Appl. Math., 49 (1996), 365-397. doi: 10.1002/(SICI)1097-0312(199604)49:4<365::AID-CPA3>3.0.CO;2-A.  Google Scholar

[6]

Bull. Amer. Math. Soc., 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[7]

Comm. Partial Differential Equations, 25 (2000), 1997-2053. doi: 10.1080/03605300008821576.  Google Scholar

[8]

Indiana Univ. Math. J., 42 (1993), 413-423. doi: 10.1512/iumj.1993.42.42019.  Google Scholar

[9]

Duke Math. J., 51 (1984), 997-1016. doi: 10.1215/S0012-7094-84-05145-7.  Google Scholar

[10]

Ph.D. Thesis, UCSB, 1996. Google Scholar

[11]

Comm. Partial Differential Equations, 23 (1998), 967-983. doi: 10.1080/03605309808821375.  Google Scholar

[12]

2nd ed., Springer-Verlag, New York, 1983.  Google Scholar

[13]

J. Differential Equations, 250 (2011), 1553-1574. doi: 10.1016/j.jde.2010.07.005.  Google Scholar

[14]

Math. Ann., 339 (2007), 461-484. doi: 10.1007/s00208-007-0125-z.  Google Scholar

[15]

J. Math. Soc. Japan, 61 (2009), 723-755. doi: 10.2969/jmsj/06130723.  Google Scholar

[16]

(Russian) Dokl. Akad. Nauk SSSR, 245 (1979), 18-20.  Google Scholar

[17]

(Russian) Izv. Akad. Nauk SSSR Ser. Mat., 44 (1980), 161-175, 239.  Google Scholar

[18]

(Russian) in "Boundary Value Problems of Mathematical Physics and Related Questions in the Theory of Functions,'' 12, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI), 96 (1980), 272-287, 312.  Google Scholar

[19]

Arch. Ration. Mech. Anal., 195 (2010), 579-607 doi: 10.1007/s00205-009-0218-9.  Google Scholar

[20]

Invent. Math., 61 (1980), 67-79. doi: 10.1007/BF01389895.  Google Scholar

[21]

Rev. Mat. Iberoamericana, 4 (1988), 453-468. doi: 10.4171/RMI/80.  Google Scholar

[22]

Comm. Pure Appl. Math., 45 (1992), 27-76. doi: 10.1002/cpa.3160450103.  Google Scholar

[23]

Z. Anal. Anwend., 28 (2009), 129-164. doi: 10.4171/ZAA/1377.  Google Scholar

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