-
Previous Article
A nonlinear eigenvalue problem for the periodic scalar $p$-Laplacian
- CPAA Home
- This Issue
-
Next Article
On some elementary properties of vector minimizers of the Allen-Cahn energy
Multi-valued solutions to a class of parabolic Monge-Ampère equations
1. | School of Mathematics and information Science, Weifang University, Shandong Weifang, 261061, China |
References:
[1] |
L. Caffarelli, Certain multiple valued harmonic functions,, \emph{Proc. Amer. Math. Soc.}, 54 (1976), 90.
|
[2] |
L. Caffarelli, On the Hölder continuity of multiple valued harmonic functions,, \emph{Indiana Univ. Math. J.}, 25 (1976), 79.
|
[3] |
L. Caffarelli, Monge-Ampère equation, Div-Curl theorems in Lagrangian coordinates, Compression and Rotation,, lecture notes., (). Google Scholar |
[4] |
L. Caffarelli and Y. Y. Li, Some multi-valued solutions to Monge-Ampère equations,, \emph{Comm. Anal. Geom.}, 14 (2006), 411.
|
[5] |
L. M. Dai and J. G. Bao, Multi-valued solutions to Hessian equations,, \emph{Nonlinear Differential Equations Appl.}, 18 (2011), 447.
doi: 10.1007/s00030-011-0103-8. |
[6] |
L. M. Dai, Multi-valued solutions to Hessian quotient equations,, \emph{Commun. Math. Sci.}, 10 (2012), 717.
doi: 10.4310/CMS.2012.v10.n2.a14. |
[7] |
G. C. Evans, A necessary and sufficient condition of Wiener,, \emph{Amer. Math. Monthly}, 54 (1947), 151.
|
[8] |
G. C. Evans, Surfaces of minimal capacity,, \emph{Proc. Nat. Acad. Sci. U. S. A.}, 26 (1940), 489.
|
[9] |
G. C. Evans, Lectures on multiple valued harmonic functions in space,, \emph{Univ. California Publ. Math. (N.S.)}, 1 (1951), 281.
|
[10] |
W. J. Firey, Shapes of worn stones,, \emph{Mathematika}, 21 (1974), 1.
|
[11] |
C. E. Gutiérrez and Q. B. Huang, $W^{2,p}$ estimates for the parabolic Monge-Ampère equation,, \emph{Arch. Ration. Mech. Anal.}, 159 (2001), 137.
doi: 10.1007/s002050100151. |
[12] |
C. E. Gutiérrez and Q. B. Huang, A generalization of a theorem by Calabi to the parabolic Monge-Ampère equation,, \emph{Indiana Univ. Math. J.}, 47 (1998), 1459.
doi: 10.1512/iumj.1998.47.1563. |
[13] |
C. E. Gutiérrez and Q. B. Huang, Geometric properties of the sections of solutions to the Monge-Ampère equation,, \emph{Trans. Amer. Math. Soc.}, 352 (2000), 4381.
doi: 10.1090/S0002-9947-00-02491-0. |
[14] |
G. Levi, Generalization of a spatial angle theorem,, (Russian) \emph{Translated from the English by Ju. V. Egorov, 26 (1971), 199.
|
[15] |
G. M. Lieberman, Second Order Parabolic Differential Equations,, World Scientific Publishing Co., (1996).
|
[16] |
N. V. Krylov, Sequences of convex functions, and estimates of the maximum of the solution of a parabolic equation,, (Russian) \emph{Sibirsk. Mat. $\breveZ$}, 17 (1976), 290.
|
[17] |
K. Tso, Deforming a hypersurface by its Gauss-Kronecker curvature,, \emph{Comm. Pure Appl. Math.}, 38 (1985), 867.
doi: 10.1002/cpa.3160380615. |
[18] |
K. Tso, On an Aleksandrov-Bakelman type maximum principle for second-order parabolic equations,, \emph{Comm. Partial Differential Equations}, 10 (1985), 543.
doi: 10.1080/03605308508820388. |
[19] |
R. H. Wang and G. L. Wang, On the existence, uniqueness and regularity of viscosity solution for the first initial boundary value problem to parabolic Monge-Ampère equations,, \emph{Northeast Math. J.}, 8 (1992), 417.
|
[20] |
R. H. Wang and G. L. Wang, The geometric measure theoretical characterization of viscosity solutions to parabolic Monge-Ampère type equation,, \emph{J. Partial Differential Equations}, 6 (1993), 237.
|
[21] |
J. G. Xiong and J. G. Bao, On Jörgens, Calabi, and Pogorelov type theorem and isolated singularities of parabolic Monge-Ampère equations,, \emph{J. Differential Equations}, 250 (2011), 367.
doi: 10.1016/j.jde.2010.08.024. |
[22] |
Y. Zhan, Viscosity Solutions of Nonlinear Degenerate Parabolic Equations and Several Applications,, Ph.D thesis, (2000).
|
show all references
References:
[1] |
L. Caffarelli, Certain multiple valued harmonic functions,, \emph{Proc. Amer. Math. Soc.}, 54 (1976), 90.
|
[2] |
L. Caffarelli, On the Hölder continuity of multiple valued harmonic functions,, \emph{Indiana Univ. Math. J.}, 25 (1976), 79.
|
[3] |
L. Caffarelli, Monge-Ampère equation, Div-Curl theorems in Lagrangian coordinates, Compression and Rotation,, lecture notes., (). Google Scholar |
[4] |
L. Caffarelli and Y. Y. Li, Some multi-valued solutions to Monge-Ampère equations,, \emph{Comm. Anal. Geom.}, 14 (2006), 411.
|
[5] |
L. M. Dai and J. G. Bao, Multi-valued solutions to Hessian equations,, \emph{Nonlinear Differential Equations Appl.}, 18 (2011), 447.
doi: 10.1007/s00030-011-0103-8. |
[6] |
L. M. Dai, Multi-valued solutions to Hessian quotient equations,, \emph{Commun. Math. Sci.}, 10 (2012), 717.
doi: 10.4310/CMS.2012.v10.n2.a14. |
[7] |
G. C. Evans, A necessary and sufficient condition of Wiener,, \emph{Amer. Math. Monthly}, 54 (1947), 151.
|
[8] |
G. C. Evans, Surfaces of minimal capacity,, \emph{Proc. Nat. Acad. Sci. U. S. A.}, 26 (1940), 489.
|
[9] |
G. C. Evans, Lectures on multiple valued harmonic functions in space,, \emph{Univ. California Publ. Math. (N.S.)}, 1 (1951), 281.
|
[10] |
W. J. Firey, Shapes of worn stones,, \emph{Mathematika}, 21 (1974), 1.
|
[11] |
C. E. Gutiérrez and Q. B. Huang, $W^{2,p}$ estimates for the parabolic Monge-Ampère equation,, \emph{Arch. Ration. Mech. Anal.}, 159 (2001), 137.
doi: 10.1007/s002050100151. |
[12] |
C. E. Gutiérrez and Q. B. Huang, A generalization of a theorem by Calabi to the parabolic Monge-Ampère equation,, \emph{Indiana Univ. Math. J.}, 47 (1998), 1459.
doi: 10.1512/iumj.1998.47.1563. |
[13] |
C. E. Gutiérrez and Q. B. Huang, Geometric properties of the sections of solutions to the Monge-Ampère equation,, \emph{Trans. Amer. Math. Soc.}, 352 (2000), 4381.
doi: 10.1090/S0002-9947-00-02491-0. |
[14] |
G. Levi, Generalization of a spatial angle theorem,, (Russian) \emph{Translated from the English by Ju. V. Egorov, 26 (1971), 199.
|
[15] |
G. M. Lieberman, Second Order Parabolic Differential Equations,, World Scientific Publishing Co., (1996).
|
[16] |
N. V. Krylov, Sequences of convex functions, and estimates of the maximum of the solution of a parabolic equation,, (Russian) \emph{Sibirsk. Mat. $\breveZ$}, 17 (1976), 290.
|
[17] |
K. Tso, Deforming a hypersurface by its Gauss-Kronecker curvature,, \emph{Comm. Pure Appl. Math.}, 38 (1985), 867.
doi: 10.1002/cpa.3160380615. |
[18] |
K. Tso, On an Aleksandrov-Bakelman type maximum principle for second-order parabolic equations,, \emph{Comm. Partial Differential Equations}, 10 (1985), 543.
doi: 10.1080/03605308508820388. |
[19] |
R. H. Wang and G. L. Wang, On the existence, uniqueness and regularity of viscosity solution for the first initial boundary value problem to parabolic Monge-Ampère equations,, \emph{Northeast Math. J.}, 8 (1992), 417.
|
[20] |
R. H. Wang and G. L. Wang, The geometric measure theoretical characterization of viscosity solutions to parabolic Monge-Ampère type equation,, \emph{J. Partial Differential Equations}, 6 (1993), 237.
|
[21] |
J. G. Xiong and J. G. Bao, On Jörgens, Calabi, and Pogorelov type theorem and isolated singularities of parabolic Monge-Ampère equations,, \emph{J. Differential Equations}, 250 (2011), 367.
doi: 10.1016/j.jde.2010.08.024. |
[22] |
Y. Zhan, Viscosity Solutions of Nonlinear Degenerate Parabolic Equations and Several Applications,, Ph.D thesis, (2000).
|
[1] |
Juhua Shi, Feida Jiang. The degenerate Monge-Ampère equations with the Neumann condition. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020297 |
[2] |
Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (1) : 301-317. doi: 10.3934/cpaa.2020267 |
[3] |
Pierre Baras. A generalization of a criterion for the existence of solutions to semilinear elliptic equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 465-504. doi: 10.3934/dcdss.2020439 |
[4] |
Tuoc Phan, Grozdena Todorova, Borislav Yordanov. Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1071-1099. doi: 10.3934/dcds.2020310 |
[5] |
Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020320 |
[6] |
Rim Bourguiba, Rosana Rodríguez-López. Existence results for fractional differential equations in presence of upper and lower solutions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1723-1747. doi: 10.3934/dcdsb.2020180 |
[7] |
Daniele Bartolucci, Changfeng Gui, Yeyao Hu, Aleks Jevnikar, Wen Yang. Mean field equations on tori: Existence and uniqueness of evenly symmetric blow-up solutions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3093-3116. doi: 10.3934/dcds.2020039 |
[8] |
Michiel Bertsch, Flavia Smarrazzo, Andrea Terracina, Alberto Tesei. Signed Radon measure-valued solutions of flux saturated scalar conservation laws. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3143-3169. doi: 10.3934/dcds.2020041 |
[9] |
Juliana Fernandes, Liliane Maia. Blow-up and bounded solutions for a semilinear parabolic problem in a saturable medium. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1297-1318. doi: 10.3934/dcds.2020318 |
[10] |
Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163 |
[11] |
Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364 |
[12] |
Hongyu Cheng, Shimin Wang. Response solutions to harmonic oscillators beyond multi–dimensional brjuno frequency. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020222 |
[13] |
Helmut Abels, Johannes Kampmann. Existence of weak solutions for a sharp interface model for phase separation on biological membranes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 331-351. doi: 10.3934/dcdss.2020325 |
[14] |
Tong Yang, Seiji Ukai, Huijiang Zhao. Stationary solutions to the exterior problems for the Boltzmann equation, I. Existence. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 495-520. doi: 10.3934/dcds.2009.23.495 |
[15] |
Ryuji Kajikiya. Existence of nodal solutions for the sublinear Moore-Nehari differential equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1483-1506. doi: 10.3934/dcds.2020326 |
[16] |
Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252 |
[17] |
Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272 |
[18] |
José Luiz Boldrini, Jonathan Bravo-Olivares, Eduardo Notte-Cuello, Marko A. Rojas-Medar. Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations. Electronic Research Archive, 2021, 29 (1) : 1783-1801. doi: 10.3934/era.2020091 |
[19] |
Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020436 |
[20] |
Dorothee Knees, Chiara Zanini. Existence of parameterized BV-solutions for rate-independent systems with discontinuous loads. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 121-149. doi: 10.3934/dcdss.2020332 |
2019 Impact Factor: 1.105
Tools
Metrics
Other articles
by authors
[Back to Top]