zbMATH — the first resource for mathematics

Existence of the global solution for the parabolic Monge-Ampère equations on compact Riemannian manifolds. (English) Zbl 1237.58027
Summary: On a compact Riemannian manifold \((M,g)\), we consider the existence and nonexistence of global solutions for the parabolic Monge-Ampère equation \[ \begin{cases}\frac{\partial}{\partial t}\varphi=\log\left(\frac{\text{det}(g+\text{Hess}\,\varphi)}{\text{det}\,g}\right)-\lambda\varphi^p-f(x),\\ \varphi(x,0)=\varphi_0(x).\end{cases}\tag{*} \] Here \(p>1\) and \(\lambda\) are real parameters. \(-f,\varphi_0:M\to(0,+\infty)\) are smooth functions on \(M\). If \(\lambda>0\), then the solution \(\varphi\) of (*) exists for all times \(t\) and \(\varphi_t=\varphi(\cdot ,t)\) converges exponentially towards a solution \(\varphi_\infty\) of its stationary equation as \(t\to\infty\). In the case of \(\lambda<0\), global solution of (*) does not exist. Thus we obtain the nonexistence of the positive solution for the stationary equation of (*).

58J35 Heat and other parabolic equation methods for PDEs on manifolds
35K96 Parabolic Monge-Ampère equations
Full Text: DOI
[1] Huisken, B., Parabolic Monge-Ampère equations on Riemannian manifolds, J. funct. anal., 147, 140-163, (1997) · Zbl 0895.58053
[2] Cao, H.D., Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds, Invent. math., 81, 359-372, (1985) · Zbl 0574.53042
[3] Guan, B.; Li, Y.Y., Monge-Ampère equations on Riemannian manifolds, J. differential equations, 132, 126-139, (1996) · Zbl 0866.58067
[4] Caffarelli, L.; Li, Y.Y., A Liouville theorem for solution of the Monge-Ampère equation with periodic data, Ann. inst. H. poincare anal. non lineaire, 21, 97-120, (2004) · Zbl 1108.35051
[5] Huang, Q., On the mean oscillation of the Hessian of solutions to the Monge-Ampère equation, Adv. math., 207, 599-616, (2006) · Zbl 1168.35356
[6] Aubin, T., Nonlinear analysis on manifolds, Monge-Ampère equations, (1982), Springer-Verlag New York · Zbl 0512.53044
[7] Schulz, F., Regularity theory for quasilinear elliptic systems and Monge-Ampère equations in two dimensions, (1982), Springer-Verlag Berlin/Heidelberg
[8] Hamilton, R.S., Three-manifolds with positive Ricci curvature, J. differential geom., 17, 255-306, (1982) · Zbl 0504.53034
[9] Oliker, V.I., Evolution of nonparametric surfaces with speed depending on curvature, I. the Gauss curvature case, Indiana univ. math. J., 40, 237-258, (1991) · Zbl 0737.53002
[10] Yau, S.T., On the Ricci curvature of compact Kähler manifold and the complex Monge-Ampère equations, I, Comm. pure appl. math., 31, 339-411, (1978) · Zbl 0369.53059
[11] Li, Y.Y., Some existence results of fully nonlinear elliptic equation of Monge-Ampère type, Comm. pure appl. math., 43, 233-271, (1990) · Zbl 0705.35038
[12] Ca, L.; Ni, L.; Spruck, J., Nonlinear second-order elliptic equations V: the Dirichlet problem for Weingarten hypersurfaces, Comm. pure appl. math., 41, 47-70, (1988) · Zbl 0672.35028
[13] Cababi, E., Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens, Michigan math. J., 5, 105-126, (1958) · Zbl 0113.30104
[14] Cheng, S.Y.; Yau, S.T., On the regularity of the solutions of the n-dimensional Minkowski problem, Comm. pure appl. math., 29, 495-516, (1976) · Zbl 0363.53030
[15] Delanoë, P., Équations du type Monge-Ampère sur LES variétés riemanniennes compactes I, J. funct. anal., 40, 358-386, (1980) · Zbl 0466.58029
[16] Delanoë, P., Équations du type Monge-Ampère sur LES variétés riemanniennes compactes II, J. funct. anal., 41, 341-353, (1981) · Zbl 0474.58023
[17] Delanoë, P., Équations du type Monge-Ampère sur LES variétés riemanniennes compactes III, J. funct. anal., 45, 403-430, (1982) · Zbl 0497.58026
[18] Trudinger, N.S., The Dirichlet problem for the prescribed curvature equations, Arch. ration. mech. anal., 111, 153-179, (1990) · Zbl 0721.35018
[19] Guan, B.; Spruck, J., Boundary value problem on \(S^n\) for surfaces of constant Gauss curvature, Ann. of math., 138, 601-624, (1993) · Zbl 0840.53046
[20] Songzhe, L., Existence of solution to initial value problem for a parabolic Monge-Ampère equation and application, Nonlinear anal., 65, 59-78, (2006) · Zbl 1100.35047
[21] Trudinger, N.S.; Wang, X.J., Boundary regularity for the Monge-Ampère and affine maximal surface equations, Ann. of math., 167, 993-1028, (2008) · Zbl 1176.35046
[22] Bakelman, I.J., Convex analysis on manifolds, Monge-Ampère equations, (1980), Springer-Verlag Berlin/Heidelberg · Zbl 0721.35017
[23] Ca, L.; Ni, L.; Spruck, J., The Dirichlet problem for nonlinear seconder-order elliptic equations I: Monge-Ampère equations, Comm. pure appl. math., 37, 339-402, (1984)
[24] Ca, L.; Ni, L.; Spruck, J., The Dirichlet problem for nonlinear seconder-order elliptic equations II: complex Monge-Ampère and uniformly elliptic equations, Comm. pure appl. math., 38, 209-252, (1985) · Zbl 0598.35048
[25] Ca, L.; Ni, L.; Spruck, J., The Dirichlet problem for nonlinear seconder-order elliptic equations III: functions of the eigenvalues of the Hessian, Acta math., 155, 261-301, (1985) · Zbl 0654.35031
[26] Xiang, N.; Yang, X.P., The complex Monge-Ampère equation with infinite boundary value, Nonlinear anal., 68, 1075-1081, (2008) · Zbl 1139.32018
[27] Trudinger, N.S.; Urbas, J.I.E., On second derivative estimates for equations of Monge-Ampère type, Bull. aust. math. soc., 30, 321-332, (1984) · Zbl 0557.35054
[28] Zhang, Z.; Wang, K., Existence and non-existence of solutions for a class of Monge-Ampère equations, J. differential equations, 246, 2849-2875, (2009) · Zbl 1165.35023
[29] Mohammed, A., Existence and estimates of solutions to a singular Dirichlet problem for the Monge-Ampère equation, J. math. anal. appl., 340, 1226-1234, (2008) · Zbl 1260.35071
[30] Gilbarg, D.; Trudinger, N.S., Elliptic partial differential equations of second-order, (1983), Springer-Verlag New York · Zbl 0691.35001
[31] Krylov, N.V., Nonlinear elliptic and parabolic equations of the second order, (1984), Reidel Boston, MA · Zbl 0578.35024
[32] Protter, M.H.; Weinberger, H.F., Maximum principles in differential equations, (1994), Springer-Verlag New York · Zbl 0153.13602
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.