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

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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

Abstract
Related Papers

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

References:

[1]	M. E. Amendola, L. Rossi and A. Vitolo, Harnack inequalities and ABP estimates for nonlinear second-order elliptic equations in unbounded domains, Abstr. Appl. Anal., (2008), Art. ID 178534, 19 pp.
[2]	X. Cabré, On the Alexandroff-Bakelman-Pucci estimate and the reversed Hölder inequality for solutions of elliptic and parabolic equations, Comm. Pure Appl. Math., 48 (1995), 539-570. doi: 10.1002/cpa.3160480504.
[3]	L. A. Caffarelli, Interior a priori estimates for solutions of fully non-linear equations, Ann. Math., 130 (1989), 189-213. doi: 10.2307/1971480.
[4]	L. A. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations,'' American Mathematical Society, Providence, 1995.
[5]	L. A. Caffarelli, M. G. Crandall, M. Kocan and A. Świech, On viscosity solutions of fully nonlinear equations with measurable ingredients, 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.
[6]	M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.
[7]	M. G. Crandall, M. Kocan and A. Świech, $L^p$-Theory for fully nonlinear uniformly parabolic equations, Comm. Partial Differential Equations, 25 (2000), 1997-2053. doi: 10.1080/03605300008821576.
[8]	L. Escauriaza, $W^{2, n}$ a priori estimates for solutions to fully non-linear equations, Indiana Univ. Math. J., 42 (1993), 413-423. doi: 10.1512/iumj.1993.42.42019.
[9]	E. B. Fabes and D. W. Stroock, The $L^p$-integrability of Green's functions and fundamental solutions for elliptic and parabolic equations, Duke Math. J., 51 (1984), 997-1016. doi: 10.1215/S0012-7094-84-05145-7.
[10]	P. K. Fok, "Some Maximum Principles and Continuity Estimates for Fully Nonlinear Elliptic Equations of Second Order,'' Ph.D. Thesis, UCSB, 1996.
[11]	K. Fok, A nonlinear Fabes-Stroock result, Comm. Partial Differential Equations, 23 (1998), 967-983. doi: 10.1080/03605309808821375.
[12]	D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,'' 2nd ed., Springer-Verlag, New York, 1983.
[13]	C. Imbert, Alexandroff-Bakelman-Pucci estimate and Harnack inequality for degenerate/singular fully non-linear elliptic equations, J. Differential Equations, 250 (2011), 1553-1574. doi: 10.1016/j.jde.2010.07.005.
[14]	S. Koike and A. Świech, Maximum principle for fully nonlinear equations via the iterated comparison function method, Math. Ann., 339 (2007), 461-484. doi: 10.1007/s00208-007-0125-z.
[15]	S. Koike and A. Świech, Weak Harnack inequality for fully nonlinear uniformly elliptic PDE with unbounded ingredients, J. Math. Soc. Japan, 61 (2009), 723-755. doi: 10.2969/jmsj/06130723.
[16]	N. V. Krylov and M. V. Safonov, An estimate for the probability of a diffusion process hitting a set of positive measure, (Russian) Dokl. Akad. Nauk SSSR, 245 (1979), 18-20.
[17]	N. V. Krylov, and M. V. Safonov, A property of the solutions of parabolic equations with measurable coefficients, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 44 (1980), 161-175, 239.
[18]	M. V. Safonov, Harnack's inequality for elliptic equations and Hölder property of their solutions, (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.
[19]	B. Sirakov, Solvability of uniformly elliptic fully nonlinear PDE, Arch. Ration. Mech. Anal., 195 (2010), 579-607 doi: 10.1007/s00205-009-0218-9.
[20]	N. S. Trudinger, Local estimates for subsolutions and supersolutions of general second order elliptic quasilinear equations, Invent. Math., 61 (1980), 67-79. doi: 10.1007/BF01389895.
[21]	N. S. Trudinger, Comparison principles and pointwise estimates for viscosity solutions of nonlinear elliptic equations, Rev. Mat. Iberoamericana, 4 (1988), 453-468. doi: 10.4171/RMI/80.
[22]	L. Wang, On the regularity of fully nonlinear parabolic equations: I, Comm. Pure Appl. Math., 45 (1992), 27-76. doi: 10.1002/cpa.3160450103.
[23]	N. Winter, $W^{2, p}$ and $W^{1, p}$-estimates at the boundary for solutions of fully nonlinear, uniformly elliptic equations, Z. Anal. Anwend., 28 (2009), 129-164. doi: 10.4171/ZAA/1377.

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References:

[1]	M. E. Amendola, L. Rossi and A. Vitolo, Harnack inequalities and ABP estimates for nonlinear second-order elliptic equations in unbounded domains, Abstr. Appl. Anal., (2008), Art. ID 178534, 19 pp.
[2]	X. Cabré, On the Alexandroff-Bakelman-Pucci estimate and the reversed Hölder inequality for solutions of elliptic and parabolic equations, Comm. Pure Appl. Math., 48 (1995), 539-570. doi: 10.1002/cpa.3160480504.
[3]	L. A. Caffarelli, Interior a priori estimates for solutions of fully non-linear equations, Ann. Math., 130 (1989), 189-213. doi: 10.2307/1971480.
[4]	L. A. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations,'' American Mathematical Society, Providence, 1995.
[5]	L. A. Caffarelli, M. G. Crandall, M. Kocan and A. Świech, On viscosity solutions of fully nonlinear equations with measurable ingredients, 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.
[6]	M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.
[7]	M. G. Crandall, M. Kocan and A. Świech, $L^p$-Theory for fully nonlinear uniformly parabolic equations, Comm. Partial Differential Equations, 25 (2000), 1997-2053. doi: 10.1080/03605300008821576.
[8]	L. Escauriaza, $W^{2, n}$ a priori estimates for solutions to fully non-linear equations, Indiana Univ. Math. J., 42 (1993), 413-423. doi: 10.1512/iumj.1993.42.42019.
[9]	E. B. Fabes and D. W. Stroock, The $L^p$-integrability of Green's functions and fundamental solutions for elliptic and parabolic equations, Duke Math. J., 51 (1984), 997-1016. doi: 10.1215/S0012-7094-84-05145-7.
[10]	P. K. Fok, "Some Maximum Principles and Continuity Estimates for Fully Nonlinear Elliptic Equations of Second Order,'' Ph.D. Thesis, UCSB, 1996.
[11]	K. Fok, A nonlinear Fabes-Stroock result, Comm. Partial Differential Equations, 23 (1998), 967-983. doi: 10.1080/03605309808821375.
[12]	D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,'' 2nd ed., Springer-Verlag, New York, 1983.
[13]	C. Imbert, Alexandroff-Bakelman-Pucci estimate and Harnack inequality for degenerate/singular fully non-linear elliptic equations, J. Differential Equations, 250 (2011), 1553-1574. doi: 10.1016/j.jde.2010.07.005.
[14]	S. Koike and A. Świech, Maximum principle for fully nonlinear equations via the iterated comparison function method, Math. Ann., 339 (2007), 461-484. doi: 10.1007/s00208-007-0125-z.
[15]	S. Koike and A. Świech, Weak Harnack inequality for fully nonlinear uniformly elliptic PDE with unbounded ingredients, J. Math. Soc. Japan, 61 (2009), 723-755. doi: 10.2969/jmsj/06130723.
[16]	N. V. Krylov and M. V. Safonov, An estimate for the probability of a diffusion process hitting a set of positive measure, (Russian) Dokl. Akad. Nauk SSSR, 245 (1979), 18-20.
[17]	N. V. Krylov, and M. V. Safonov, A property of the solutions of parabolic equations with measurable coefficients, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 44 (1980), 161-175, 239.
[18]	M. V. Safonov, Harnack's inequality for elliptic equations and Hölder property of their solutions, (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.
[19]	B. Sirakov, Solvability of uniformly elliptic fully nonlinear PDE, Arch. Ration. Mech. Anal., 195 (2010), 579-607 doi: 10.1007/s00205-009-0218-9.
[20]	N. S. Trudinger, Local estimates for subsolutions and supersolutions of general second order elliptic quasilinear equations, Invent. Math., 61 (1980), 67-79. doi: 10.1007/BF01389895.
[21]	N. S. Trudinger, Comparison principles and pointwise estimates for viscosity solutions of nonlinear elliptic equations, Rev. Mat. Iberoamericana, 4 (1988), 453-468. doi: 10.4171/RMI/80.
[22]	L. Wang, On the regularity of fully nonlinear parabolic equations: I, Comm. Pure Appl. Math., 45 (1992), 27-76. doi: 10.1002/cpa.3160450103.
[23]	N. Winter, $W^{2, p}$ and $W^{1, p}$-estimates at the boundary for solutions of fully nonlinear, uniformly elliptic equations, Z. Anal. Anwend., 28 (2009), 129-164. doi: 10.4171/ZAA/1377.

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