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

show all references

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

[1]

Francesca Da Lio. Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 395-415. doi: 10.3934/cpaa.2004.3.395

[2]

Robert Jensen, Andrzej Świech. Uniqueness and existence of maximal and minimal solutions of fully nonlinear elliptic PDE. Communications on Pure & Applied Analysis, 2005, 4 (1) : 199-207. doi: 10.3934/cpaa.2005.4.187

[3]

Rainer Buckdahn, Christian Keller, Jin Ma, Jianfeng Zhang. Fully nonlinear stochastic and rough PDEs: Classical and viscosity solutions. Probability, Uncertainty and Quantitative Risk, 2020, 5 (0) : 7-. doi: 10.1186/s41546-020-00049-8

[4]

Ibrahim Ekren, Jianfeng Zhang. Pseudo-Markovian viscosity solutions of fully nonlinear degenerate PPDEs. Probability, Uncertainty and Quantitative Risk, 2016, 1 (0) : 6-. doi: 10.1186/s41546-016-0010-3

[5]

Isabeau Birindelli, Francoise Demengel. Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators. Communications on Pure & Applied Analysis, 2007, 6 (2) : 335-366. doi: 10.3934/cpaa.2007.6.335

[6]

Limei Dai. Entire solutions with asymptotic behavior of fully nonlinear uniformly elliptic equations. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1707-1714. doi: 10.3934/cpaa.2011.10.1707

[7]

Luisa Fattorusso, Antonio Tarsia. Regularity in Campanato spaces for solutions of fully nonlinear elliptic systems. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1307-1323. doi: 10.3934/dcds.2011.31.1307

[8]

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

[9]

Antonio Vitolo, Maria E. Amendola, Giulio Galise. On the uniqueness of blow-up solutions of fully nonlinear elliptic equations. Conference Publications, 2013, 2013 (special) : 771-780. doi: 10.3934/proc.2013.2013.771

[10]

Chuanqiang Chen. On the microscopic spacetime convexity principle for fully nonlinear parabolic equations II: Spacetime quasiconcave solutions. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 4761-4811. doi: 10.3934/dcds.2016007

[11]

Chuanqiang Chen. On the microscopic spacetime convexity principle of fully nonlinear parabolic equations I: Spacetime convex solutions. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 3383-3402. doi: 10.3934/dcds.2014.34.3383

[12]

Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Discrete & Continuous Dynamical Systems, 2008, 21 (3) : 763-800. doi: 10.3934/dcds.2008.21.763

[13]

Chiun-Chuan Chen, Li-Chang Hung, Hsiao-Feng Liu. N-barrier maximum principle for degenerate elliptic systems and its application. Discrete & Continuous Dynamical Systems, 2018, 38 (2) : 791-821. doi: 10.3934/dcds.2018034

[14]

Tomasz Komorowski, Adam Bobrowski. A quantitative Hopf-type maximum principle for subsolutions of elliptic PDEs. Discrete & Continuous Dynamical Systems - S, 2020, 13 (12) : 3495-3502. doi: 10.3934/dcdss.2020248

[15]

Xiaoming He, Xin Zhao, Wenming Zou. Maximum principles for a fully nonlinear nonlocal equation on unbounded domains. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4387-4399. doi: 10.3934/cpaa.2020200

[16]

Wenmin Sun, Jiguang Bao. New maximum principles for fully nonlinear ODEs of second order. Discrete & Continuous Dynamical Systems, 2007, 19 (4) : 813-823. doi: 10.3934/dcds.2007.19.813

[17]

Italo Capuzzo Dolcetta, Antonio Vitolo. Glaeser's type gradient estimates for non-negative solutions of fully nonlinear elliptic equations. Discrete & Continuous Dynamical Systems, 2010, 28 (2) : 539-557. doi: 10.3934/dcds.2010.28.539

[18]

Luca Rossi. Non-existence of positive solutions of fully nonlinear elliptic equations in unbounded domains. Communications on Pure & Applied Analysis, 2008, 7 (1) : 125-141. doi: 10.3934/cpaa.2008.7.125

[19]

H. O. Fattorini. The maximum principle in infinite dimension. Discrete & Continuous Dynamical Systems, 2000, 6 (3) : 557-574. doi: 10.3934/dcds.2000.6.557

[20]

Adel Chala, Dahbia Hafayed. On stochastic maximum principle for risk-sensitive of fully coupled forward-backward stochastic control of mean-field type with application. Evolution Equations & Control Theory, 2020, 9 (3) : 817-843. doi: 10.3934/eect.2020035

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (122)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]