# American Institute of Mathematical Sciences

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

 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.
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] 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] L. A. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations,'' American Mathematical Society, Providence, 1995.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] P. K. Fok, "Some Maximum Principles and Continuity Estimates for Fully Nonlinear Elliptic Equations of Second Order,'' Ph.D. Thesis, UCSB, 1996. Google Scholar [11] K. Fok, A nonlinear Fabes-Stroock result, Comm. Partial Differential Equations, 23 (1998), 967-983. doi: 10.1080/03605309808821375.  Google Scholar [12] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,'' 2nd ed., Springer-Verlag, New York, 1983.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] L. A. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations,'' American Mathematical Society, Providence, 1995.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] P. K. Fok, "Some Maximum Principles and Continuity Estimates for Fully Nonlinear Elliptic Equations of Second Order,'' Ph.D. Thesis, UCSB, 1996. Google Scholar [11] K. Fok, A nonlinear Fabes-Stroock result, Comm. Partial Differential Equations, 23 (1998), 967-983. doi: 10.1080/03605309808821375.  Google Scholar [12] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,'' 2nd ed., Springer-Verlag, New York, 1983.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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