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Existence and multiplicity for Hamilton-Jacobi-Bellman equation
1. | College of Science, Northwest A & F University, Yangling, Shaanxi 712100, China |
2. | School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China |
$ \begin{equation*} \left\{ \begin{array}{l} -\mathcal{M}_\mathcal{C}^{\pm}(D^2u) = \mu f(u) \ \ \ \ \text{in} \ \ \Omega,\\ u = 0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{on}\ \partial\Omega, \end{array} \right. \end{equation*} $ |
$ \Omega\subset\mathbb{R}^N $ |
$ N\geq3 $ |
$ \mathcal{M}_\mathcal{C}^{\pm} $ |
$ \mu $ |
$ \mu $ |
$ f $ |
$ 0 $ |
$ \infty $ |
$ f $ |
$ f(s)s>0 $ |
$ s\neq0 $ |
References:
[1] |
S. Alarcón, L. Iturriaga and A. Quaas,
Existence and multiplicity results for Pucci's operators involving nonlinearities with zeros, Calc. Var. Partial Differ. Equ., 45 (2012), 443-454.
doi: 10.1007/s00526-011-0465-0. |
[2] |
S. N. Armstrong,
Principal eigenvalues and an anti-maximum principle for homogeneous fully nonlinear elliptic equations, J. Differ. Equ., 246 (2009), 2958-2987.
doi: 10.1016/j.jde.2008.10.026. |
[3] |
R. Bellman, Dynamic Programming, Princeton Univ. Press, Princeton, NJ., 1957.
![]() ![]() |
[4] |
I. Birindelli and F. Demengel, Comparison principle and Liouville type results for singular fully nonlinear operators, Ann. Fac. Sci. Toulouse Math., 13 (2004), 261-287.
doi: 10.5802/afst.1070. |
[5] |
J. Busca, M. J. Esteban and A. Quaas,
Nonlinear eigenvalues and bifurcation problems for Pucci's operators, Ann. I. H. Poincaré-AN, 22 (2005), 187-206.
doi: 10.1016/j.anihpc.2004.05.004. |
[6] |
A. Bensoussan and J. L. Lions, Applications of Variational Inequalities in Stochastic Control., Translated from French, In: ''Studies in Mathematics and its Applications" 12, North-Holland Publishing Co., Amsterdam New York, 1982. |
[7] |
G. W. Dai,
Bifurcation and one-sign solutions of the p-Laplacian involving a nonlinearity with zeros, Discrete Contin. Dyn. Syst., 36 (2016), 5323-5345.
doi: 10.3934/dcds.2016034. |
[8] |
G. W. Dai, Two Whyburn type topological theorems and its applications to Monge-Ampère equations, Calc. Var. Partial Differ. Equ., 55 (2016), 97pp.
doi: 10.1007/s00526-016-1029-0. |
[9] |
G. W. Dai,
Generalized limit theorem and bifurcation for problems with Pucci's operator, Topol. Methods Nonlinear Anal., 56 (2020), 229-261.
doi: 10.12775/TMNA.2020.012. |
[10] |
P. L. Felmer and A. Quaas,
Fundamental solutions and two properties of elliptic maximal and minimal operators, Trans. Amer. Math. Soc., 361 (11) (2009), 5721-5736.
doi: 10.1090/S0002-9947-09-04566-8. |
[11] |
B. Gidas and J. Spruck,
A priori bounds of positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901.
doi: 10.1080/03605308108820196. |
[12] |
Y. X. Hua and X. H. Yu,
Liouville type theorem and decay estimates for solutions of fully nonlinear elliptic equation, J. Math. Anal. Appl., 405 (2013), 608-617.
doi: 10.1016/j.jmaa.2013.04.025. |
[13] |
H. J. Kappen, Optimal control theory and the linear Bellman Equation, Inference and Learning in Dynamic Models, (2011), 363–387.
doi: https://doi.org/10.1017/CBO9780511984679.018. |
[14] |
P. L. Lions, Generalized Solutions of Hamilton-Jacobi Equations, Pitman, London, 1982. |
[15] |
P. L. Lions,
Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. I. The dynamic programming principle and applications, Commun. Partial Differ. Equ., 8 (1983), 1101-1174.
doi: 10.1080/03605308308820297. |
[16] |
P. L. Lions,
Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. II. Viscosity solutions and uniqueness, Commun. Partial Differ. Equ., 8 (1983), 1229-1276.
doi: 10.1080/03605308308820301. |
[17] |
P. L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. III. Regularity of the optimal cost function, in: Nonlinear Partial Differential Equations and Their Applications, Collège de France seminar, V (1983), 95–205.
doi: https://doi.org/10.1017/CBO9780511984679.018. |
[18] |
P. L. Lions,
Bifurcation and optimal stochastic control, Nonlinear Anal., 2 (1983), 177-207.
doi: 10.1016/0362-546X(83)90081-0. |
[19] |
C. Pucci,
Maximum and minimum first eigenvalues for a class of elliptic operators, Proc. Amer. Math. Soc., 17 (1966), 788-795.
doi: 10.2307/2036253. |
[20] |
C. Pucci,
Operatori ellittici estremanti, Ann. Mat. Pure Appl., 72 (1966), 141-170.
doi: 10.1007/BF02414332. |
[21] |
A. Quaas, Existence of a positive solution to a "semilinear" equation involving Pucci's operator in a convex domain, Differ. Integral Equ., 17 (2004), 481-494. |
[22] |
A. Quaas and B. Sirakov,
Existence results for nonproper elliptic equations involving the Pucci's operator, Commun. Partial Differ. Equ., 31 (2006), 987-1003.
doi: 10.1080/03605300500394421. |
[23] |
A. Quaas and B. Sirakov,
Principal eigenvalues and the Dirichlet problem for fully nonlinear operators, Adv. Math., 218 (2008), 105-135.
doi: 10.1016/j.aim.2007.12.002. |
[24] |
A. Quaas and A. Allendes,
Multiplicity results for extremal operators through bifurcation, Discrete Contin. Dyn. Syst., 29 (2011), 51-65.
doi: 10.3934/dcds.2011.29.51. |
[25] |
X. H. Yu,
Multiplicity solutions for fully nonlinear equation involving nonlinearity with zeros, Comm. Pure Appl. Math., 12 (2013), 451-459.
doi: 10.3934/cpaa.2013.12.451. |
show all references
References:
[1] |
S. Alarcón, L. Iturriaga and A. Quaas,
Existence and multiplicity results for Pucci's operators involving nonlinearities with zeros, Calc. Var. Partial Differ. Equ., 45 (2012), 443-454.
doi: 10.1007/s00526-011-0465-0. |
[2] |
S. N. Armstrong,
Principal eigenvalues and an anti-maximum principle for homogeneous fully nonlinear elliptic equations, J. Differ. Equ., 246 (2009), 2958-2987.
doi: 10.1016/j.jde.2008.10.026. |
[3] |
R. Bellman, Dynamic Programming, Princeton Univ. Press, Princeton, NJ., 1957.
![]() ![]() |
[4] |
I. Birindelli and F. Demengel, Comparison principle and Liouville type results for singular fully nonlinear operators, Ann. Fac. Sci. Toulouse Math., 13 (2004), 261-287.
doi: 10.5802/afst.1070. |
[5] |
J. Busca, M. J. Esteban and A. Quaas,
Nonlinear eigenvalues and bifurcation problems for Pucci's operators, Ann. I. H. Poincaré-AN, 22 (2005), 187-206.
doi: 10.1016/j.anihpc.2004.05.004. |
[6] |
A. Bensoussan and J. L. Lions, Applications of Variational Inequalities in Stochastic Control., Translated from French, In: ''Studies in Mathematics and its Applications" 12, North-Holland Publishing Co., Amsterdam New York, 1982. |
[7] |
G. W. Dai,
Bifurcation and one-sign solutions of the p-Laplacian involving a nonlinearity with zeros, Discrete Contin. Dyn. Syst., 36 (2016), 5323-5345.
doi: 10.3934/dcds.2016034. |
[8] |
G. W. Dai, Two Whyburn type topological theorems and its applications to Monge-Ampère equations, Calc. Var. Partial Differ. Equ., 55 (2016), 97pp.
doi: 10.1007/s00526-016-1029-0. |
[9] |
G. W. Dai,
Generalized limit theorem and bifurcation for problems with Pucci's operator, Topol. Methods Nonlinear Anal., 56 (2020), 229-261.
doi: 10.12775/TMNA.2020.012. |
[10] |
P. L. Felmer and A. Quaas,
Fundamental solutions and two properties of elliptic maximal and minimal operators, Trans. Amer. Math. Soc., 361 (11) (2009), 5721-5736.
doi: 10.1090/S0002-9947-09-04566-8. |
[11] |
B. Gidas and J. Spruck,
A priori bounds of positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901.
doi: 10.1080/03605308108820196. |
[12] |
Y. X. Hua and X. H. Yu,
Liouville type theorem and decay estimates for solutions of fully nonlinear elliptic equation, J. Math. Anal. Appl., 405 (2013), 608-617.
doi: 10.1016/j.jmaa.2013.04.025. |
[13] |
H. J. Kappen, Optimal control theory and the linear Bellman Equation, Inference and Learning in Dynamic Models, (2011), 363–387.
doi: https://doi.org/10.1017/CBO9780511984679.018. |
[14] |
P. L. Lions, Generalized Solutions of Hamilton-Jacobi Equations, Pitman, London, 1982. |
[15] |
P. L. Lions,
Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. I. The dynamic programming principle and applications, Commun. Partial Differ. Equ., 8 (1983), 1101-1174.
doi: 10.1080/03605308308820297. |
[16] |
P. L. Lions,
Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. II. Viscosity solutions and uniqueness, Commun. Partial Differ. Equ., 8 (1983), 1229-1276.
doi: 10.1080/03605308308820301. |
[17] |
P. L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. III. Regularity of the optimal cost function, in: Nonlinear Partial Differential Equations and Their Applications, Collège de France seminar, V (1983), 95–205.
doi: https://doi.org/10.1017/CBO9780511984679.018. |
[18] |
P. L. Lions,
Bifurcation and optimal stochastic control, Nonlinear Anal., 2 (1983), 177-207.
doi: 10.1016/0362-546X(83)90081-0. |
[19] |
C. Pucci,
Maximum and minimum first eigenvalues for a class of elliptic operators, Proc. Amer. Math. Soc., 17 (1966), 788-795.
doi: 10.2307/2036253. |
[20] |
C. Pucci,
Operatori ellittici estremanti, Ann. Mat. Pure Appl., 72 (1966), 141-170.
doi: 10.1007/BF02414332. |
[21] |
A. Quaas, Existence of a positive solution to a "semilinear" equation involving Pucci's operator in a convex domain, Differ. Integral Equ., 17 (2004), 481-494. |
[22] |
A. Quaas and B. Sirakov,
Existence results for nonproper elliptic equations involving the Pucci's operator, Commun. Partial Differ. Equ., 31 (2006), 987-1003.
doi: 10.1080/03605300500394421. |
[23] |
A. Quaas and B. Sirakov,
Principal eigenvalues and the Dirichlet problem for fully nonlinear operators, Adv. Math., 218 (2008), 105-135.
doi: 10.1016/j.aim.2007.12.002. |
[24] |
A. Quaas and A. Allendes,
Multiplicity results for extremal operators through bifurcation, Discrete Contin. Dyn. Syst., 29 (2011), 51-65.
doi: 10.3934/dcds.2011.29.51. |
[25] |
X. H. Yu,
Multiplicity solutions for fully nonlinear equation involving nonlinearity with zeros, Comm. Pure Appl. Math., 12 (2013), 451-459.
doi: 10.3934/cpaa.2013.12.451. |


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