# 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).   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.  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.  doi: 10.2307/1971480.  Google Scholar [4] L. A. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations,'', American Mathematical Society, (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.  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.  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.  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.  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.  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, (1996).   Google Scholar [11] K. Fok, A nonlinear Fabes-Stroock result,, Comm. Partial Differential Equations, 23 (1998), 967.  doi: 10.1080/03605309808821375.  Google Scholar [12] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,'', 2nd ed., (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.  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.  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.  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.   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.   Google Scholar [18] M. V. Safonov, Harnack's inequality for elliptic equations and Hölder property of their solutions,, (Russian) in, 96 (1980), 272.   Google Scholar [19] B. Sirakov, Solvability of uniformly elliptic fully nonlinear PDE,, Arch. Ration. Mech. Anal., 195 (2010), 579.  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.  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.  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.  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.  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).   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.  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.  doi: 10.2307/1971480.  Google Scholar [4] L. A. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations,'', American Mathematical Society, (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.  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.  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.  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.  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.  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, (1996).   Google Scholar [11] K. Fok, A nonlinear Fabes-Stroock result,, Comm. Partial Differential Equations, 23 (1998), 967.  doi: 10.1080/03605309808821375.  Google Scholar [12] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,'', 2nd ed., (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.  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.  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.  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.   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.   Google Scholar [18] M. V. Safonov, Harnack's inequality for elliptic equations and Hölder property of their solutions,, (Russian) in, 96 (1980), 272.   Google Scholar [19] B. Sirakov, Solvability of uniformly elliptic fully nonlinear PDE,, Arch. Ration. Mech. Anal., 195 (2010), 579.  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.  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.  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.  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.  doi: 10.4171/ZAA/1377.  Google Scholar
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