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Entire subsolutions of Monge-Ampère type equations
1. | School of Mathematics and Information Science, Weifang University, Weifang, 261061, China |
2. | College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, China |
In this paper, we consider the subsolutions of the Monge-Ampère type equations $ {\det}^{\frac{1}{n}}(D^2u+\alpha I) = f(u) $ in $ \mathbb{R}^{n} $. We obtain the necessary and sufficient condition of the existence of subsolutions.
References:
[1] |
H. Brezis,
Semilinear equations in ${\mathbb R}^N$ without condition at infinity, Appl. Math. Optim., 12 (1984), 271-282.
doi: 10.1007/BF01449045. |
[2] |
J. G. Bao, X. H. Ji and H. G. Li,
Existence and nonexistence theorem for entire subsolutions of $k$-Yamabe type equations, J. Differential Equations, 253 (2012), 2140-2160.
doi: 10.1016/j.jde.2012.06.018. |
[3] |
L. A. Caffarelli, L. Nirenberg and J. Spruck,
The Dirichlet problem for nonlinear second-order elliptic equations, I. Monge-Ampère equation, Comm. Pure Appl. Math., 37 (1984), 369-402.
doi: 10.1002/cpa.3160370306. |
[4] |
I. Capuzzo Dolcetta, F. Leoni and A. Vitolo,
Entire subsolutions of fully nonlinear degenerate elliptic equations, Bull. Inst. Math. Acad. Sin. (N.S.), 9 (2014), 147-161.
|
[5] |
P. F. Guan and X. J. Wang,
On a Monge-Ampère equation arising in geometric optics, J. Differential Geom., 48 (1998), 205-223.
|
[6] |
Q. Han and J. X. Hong, Isometric Embedding of Riemannian Manifolds in Euclidean Spaces, Mathematical Surveys and Monographs, 130, American Mathematical Society, Providence, RI, 2006.
doi: 10.1090/surv/130. |
[7] |
Y. Huang, F. D. Jiang and J. K. Liu,
Boundary $C^ {2, \alpha}$ estimates for Monge-Ampère type equations, Adv. Math., 281 (2015), 706-733.
doi: 10.1016/j.aim.2014.12.043. |
[8] |
N. M. Ivochkina, Classical solvability of the Dirichlet problem for the Monge-Ampère equation, (Russian), Questions in Quantum Field Theory and Statistical Physics, 4. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI)), 131 (1983), 72–79. |
[9] |
X. H. Ji and J. G. Bao,
Necessary and sufficient conditions on solvability for Hessian inequalities, Proc. Amer. Math. Soc., 138 (2010), 175-188.
doi: 10.1090/S0002-9939-09-10032-1. |
[10] |
F. D. Jiang, N. S. Trudinger and X. P. Yang,
On the Dirichlet problem for Monge-Ampère type equations, Calc. Var. Partial Differential Equations, 49 (2014), 1223-1236.
doi: 10.1007/s00526-013-0619-3. |
[11] |
Q. N. Jin, Y. Y. Li and H. Y. Xu,
Nonexistence of positive solutions for some fully nonlinear elliptic equations, Methods Appl. Anal., 12 (2005), 441-449.
doi: 10.4310/MAA.2005.v12.n4.a5. |
[12] |
J. B. Keller,
On solutions of $\Delta u = f(u)$, Comm. Pure Appl. Math., 10 (1957), 503-510.
doi: 10.1002/cpa.3160100402. |
[13] |
N. V. Krylov,
Boundedly inhomogeneous elliptic and parabolic equations in a domain, Izv. Akad. Nauk SSSR Ser. Mat., 47 (1983), 75-108.
|
[14] |
J. K. Liu, N. S. Trudinger and X. J. Wang,
Interior $C^{2, \alpha}$ regularity for potential functions in optimal transportation, Comm. Partial Differential Equations, 35 (2010), 165-184.
doi: 10.1080/03605300903236609. |
[15] |
C. Loewner and L. Nirenberg, Partial differential equations invariant under conformal or projective transformations, in Contributions to Analysis (a collection of papers dedicated to Lipman Bers), Academic Press, (1974), 245–272. |
[16] |
X. N. Ma, N. S. Trudinger and X. J. Wang,
Regularity of potential functions of the optimal transportation problem, Arch. Ration. Mech. Anal., 177 (2005), 151-183.
doi: 10.1007/s00205-005-0362-9. |
[17] |
R. Osserman,
On the inequality $\Delta u\geq f(u)$, Pacific J. Math., 7 (1957), 1641-1647.
|
[18] |
N. S. Trudinger and X. J. Wang, The Monge-Ampère equation and its geometric applications, Adv. Lect. Math. (Handbook of geometric analysis, a collection of papers dedicated to Lipman Bers), Int. Press, Somerville, MA, 1 (2008), 467–524. |
[19] |
N. S. Trudinger and X. J. Wang,
On the second boundary value problem for Monge-Ampère type equations and optimal transportation, Ann. Sc. Norm. Super. Pisa Cl. Sci., 8 (2009), 143-174.
|
[20] |
X. J. Wang,
On the design of a reflector antenna, Inverse Problems, 12 (1996), 351-375.
doi: 10.1088/0266-5611/12/3/013. |
show all references
References:
[1] |
H. Brezis,
Semilinear equations in ${\mathbb R}^N$ without condition at infinity, Appl. Math. Optim., 12 (1984), 271-282.
doi: 10.1007/BF01449045. |
[2] |
J. G. Bao, X. H. Ji and H. G. Li,
Existence and nonexistence theorem for entire subsolutions of $k$-Yamabe type equations, J. Differential Equations, 253 (2012), 2140-2160.
doi: 10.1016/j.jde.2012.06.018. |
[3] |
L. A. Caffarelli, L. Nirenberg and J. Spruck,
The Dirichlet problem for nonlinear second-order elliptic equations, I. Monge-Ampère equation, Comm. Pure Appl. Math., 37 (1984), 369-402.
doi: 10.1002/cpa.3160370306. |
[4] |
I. Capuzzo Dolcetta, F. Leoni and A. Vitolo,
Entire subsolutions of fully nonlinear degenerate elliptic equations, Bull. Inst. Math. Acad. Sin. (N.S.), 9 (2014), 147-161.
|
[5] |
P. F. Guan and X. J. Wang,
On a Monge-Ampère equation arising in geometric optics, J. Differential Geom., 48 (1998), 205-223.
|
[6] |
Q. Han and J. X. Hong, Isometric Embedding of Riemannian Manifolds in Euclidean Spaces, Mathematical Surveys and Monographs, 130, American Mathematical Society, Providence, RI, 2006.
doi: 10.1090/surv/130. |
[7] |
Y. Huang, F. D. Jiang and J. K. Liu,
Boundary $C^ {2, \alpha}$ estimates for Monge-Ampère type equations, Adv. Math., 281 (2015), 706-733.
doi: 10.1016/j.aim.2014.12.043. |
[8] |
N. M. Ivochkina, Classical solvability of the Dirichlet problem for the Monge-Ampère equation, (Russian), Questions in Quantum Field Theory and Statistical Physics, 4. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI)), 131 (1983), 72–79. |
[9] |
X. H. Ji and J. G. Bao,
Necessary and sufficient conditions on solvability for Hessian inequalities, Proc. Amer. Math. Soc., 138 (2010), 175-188.
doi: 10.1090/S0002-9939-09-10032-1. |
[10] |
F. D. Jiang, N. S. Trudinger and X. P. Yang,
On the Dirichlet problem for Monge-Ampère type equations, Calc. Var. Partial Differential Equations, 49 (2014), 1223-1236.
doi: 10.1007/s00526-013-0619-3. |
[11] |
Q. N. Jin, Y. Y. Li and H. Y. Xu,
Nonexistence of positive solutions for some fully nonlinear elliptic equations, Methods Appl. Anal., 12 (2005), 441-449.
doi: 10.4310/MAA.2005.v12.n4.a5. |
[12] |
J. B. Keller,
On solutions of $\Delta u = f(u)$, Comm. Pure Appl. Math., 10 (1957), 503-510.
doi: 10.1002/cpa.3160100402. |
[13] |
N. V. Krylov,
Boundedly inhomogeneous elliptic and parabolic equations in a domain, Izv. Akad. Nauk SSSR Ser. Mat., 47 (1983), 75-108.
|
[14] |
J. K. Liu, N. S. Trudinger and X. J. Wang,
Interior $C^{2, \alpha}$ regularity for potential functions in optimal transportation, Comm. Partial Differential Equations, 35 (2010), 165-184.
doi: 10.1080/03605300903236609. |
[15] |
C. Loewner and L. Nirenberg, Partial differential equations invariant under conformal or projective transformations, in Contributions to Analysis (a collection of papers dedicated to Lipman Bers), Academic Press, (1974), 245–272. |
[16] |
X. N. Ma, N. S. Trudinger and X. J. Wang,
Regularity of potential functions of the optimal transportation problem, Arch. Ration. Mech. Anal., 177 (2005), 151-183.
doi: 10.1007/s00205-005-0362-9. |
[17] |
R. Osserman,
On the inequality $\Delta u\geq f(u)$, Pacific J. Math., 7 (1957), 1641-1647.
|
[18] |
N. S. Trudinger and X. J. Wang, The Monge-Ampère equation and its geometric applications, Adv. Lect. Math. (Handbook of geometric analysis, a collection of papers dedicated to Lipman Bers), Int. Press, Somerville, MA, 1 (2008), 467–524. |
[19] |
N. S. Trudinger and X. J. Wang,
On the second boundary value problem for Monge-Ampère type equations and optimal transportation, Ann. Sc. Norm. Super. Pisa Cl. Sci., 8 (2009), 143-174.
|
[20] |
X. J. Wang,
On the design of a reflector antenna, Inverse Problems, 12 (1996), 351-375.
doi: 10.1088/0266-5611/12/3/013. |
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