黎曼流形上抛物Monge-Ampère方程解的存在性与非存在性
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摘要
本文主要考虑n维紧黎曼流形(M,g)上的抛物Monge-Ampere方程：
整体解的存在性与非存在性,其中λ和p都是实参数且p>1,-f,φ0：M→(0,+∞)都是流形M上的光滑函数,g是流形M上的黎曼度量.文章分为三个部分：第一部分主要介绍先前和当前Monge-Ampere方程的研究动态和主要结果,该部分还包括本文所做的主要工作、获得的主要结论以及与流形相关部分知识的介绍；第二部分也就是正文部分,包括解的先验估计、n+△φ的估计以及本文主要结果的证明：若λ>0,则方程(*)的整体解φ(x,t)存在,且当t→∞时,解φt=φ(·,t)收敛到其定态方程的唯一解φ∞(x)；第三部分主要考虑λ<0且f=0的情形,应用抛物方程的比较原理,获得了两个结果：(ⅰ)若方程的初值φ0(x)>0,则方程(*)的整体解不存在；(ⅱ)方程(*)的定态方程：
不存在大于零的解φ∞(x).
In this paper, we just consider the existence and non-existence of global solutions for parabolic Monge-Ampere equations on compact Riemannian Manifolds (M,g).
Here A and p> 1 are real parameters.-f/,φ0:M→(0,+∞) are smooth functions on compact Riemannian manifold (M,g) and g is a Riemannian metric on M. This article is divided into three parts:In the first part, we just introduce previous and current situation of Monge-Ampere equations as well as main results that have acquired in their papers. Of course, the work done by this paper and main results that have acquired in this article are included in this part. Prepared knowledge of Riemannian Manifolds are also introduced in the first part. The body of this article is the second part. In the second part, it includes a priori estimates of solution and n+Δφ. The main conclusions of this paper are as follows:ifλ> 0, then the solutionφof equation (*) exists for all times t andφt=φ(·,t) converges exponentially towards a solutionφ∞of its stationary equation as t→∞. In the third part, supposingλ< 0 and f= 0, two results will be obtained by applying Comparison Principle of parabolic equations to equation (*):First, ifφ0(x)> 0, then the global solution of equation (*) doesn't exist; Second, the stationary equation of equation (*):
does not have positive solutionφ∞(x).
整体解的存在性与非存在性,其中λ和p都是实参数且p>1,-f,φ0：M→(0,+∞)都是流形M上的光滑函数,g是流形M上的黎曼度量.文章分为三个部分：第一部分主要介绍先前和当前Monge-Ampere方程的研究动态和主要结果,该部分还包括本文所做的主要工作、获得的主要结论以及与流形相关部分知识的介绍；第二部分也就是正文部分,包括解的先验估计、n+△φ的估计以及本文主要结果的证明：若λ>0,则方程(*)的整体解φ(x,t)存在,且当t→∞时,解φt=φ(·,t)收敛到其定态方程的唯一解φ∞(x)；第三部分主要考虑λ<0且f=0的情形,应用抛物方程的比较原理,获得了两个结果：(ⅰ)若方程的初值φ0(x)>0,则方程(*)的整体解不存在；(ⅱ)方程(*)的定态方程：
不存在大于零的解φ∞(x).
In this paper, we just consider the existence and non-existence of global solutions for parabolic Monge-Ampere equations on compact Riemannian Manifolds (M,g).
Here A and p> 1 are real parameters.-f/,φ0:M→(0,+∞) are smooth functions on compact Riemannian manifold (M,g) and g is a Riemannian metric on M. This article is divided into three parts:In the first part, we just introduce previous and current situation of Monge-Ampere equations as well as main results that have acquired in their papers. Of course, the work done by this paper and main results that have acquired in this article are included in this part. Prepared knowledge of Riemannian Manifolds are also introduced in the first part. The body of this article is the second part. In the second part, it includes a priori estimates of solution and n+Δφ. The main conclusions of this paper are as follows:ifλ> 0, then the solutionφof equation (*) exists for all times t andφt=φ(·,t) converges exponentially towards a solutionφ∞of its stationary equation as t→∞. In the third part, supposingλ< 0 and f= 0, two results will be obtained by applying Comparison Principle of parabolic equations to equation (*):First, ifφ0(x)> 0, then the global solution of equation (*) doesn't exist; Second, the stationary equation of equation (*):
does not have positive solutionφ∞(x).
引文
[1]Zhitao Zhang and Kelei Wang, Existence and non-existence of solutions for a class of Monge-Ampere Equations, J.Differential Equations,2009,246:2849-2875.
[2]Ahmed Mohammed,Existence and estimates of solutions to a singular Dirichlet problem for the Monge-Ampere equation, J.Math.Anal.Appl.,2008,340:1226-1234.
[3]H.-D. Cao, Deformation of Kahler metrics to Kahlealer-Einstein metrics on compact kahler manifolds, Invent.Math.,1958,81:359-372.
[4]Barbara Huisken, Parabolic Monge-Ampere Equations on Riemanian Manifolds, Jour-nal of Functional Analysis,1997,147:140-163.
[5]Bo Guan and Yan Yan Li, Monge-Ampere Equations on Riemannian Manifolds, Journal Of Differential Equations,1996,132:126-139.
[6]Lian Songzhe, Existence of solution to initial value problem for a parabolic Monge-Ampere equation and application, Nonlinear Analysis,2006,65:59-78.
[7]L.Caffarelli,YanYan Li, A Liouville theorem for solution of the Monge-Ampere equation with periodic data, Ann.I.H.Poin.,2004,21:97-120.
[8]Ni Xiang,Xiao-Ping Yang, The complex Monge-Ampere equation with infinite bound-ary value, Nonlinear Analysis,2008,68:1075-1081.
[9]I.J.Bakelman, Convex Analysis on Manifolds, Monge-Ampere Equations, Springer-Verlag, Berlin/Heidelberg,1980.
[10]Qingbo Huang, On the mean oscillation of the Hessian of solutions to the Monge-Ampere equation, Advances in Mathematics,2006,207:599-616.
[11]R.Chang yu and W.Guang lie, Parabolic Monge-Ampere Equation on Riemannian Manifolds, Northeast Math.J.,2007,23(5):413-424.
[12]N.S.Trudinger and X.J.Wang, Boundary regularity for the Monge-Ampere and affine maximal surface equations, Ann.of Math.,2008,167:993-1028.
[13]A.D.Alexandrov, Dirichlets'problem for the equation Det‖Zij‖=φ, Vestnik Leningrad Univ.,1985,13, No.1:5-24.[In Russian]
[14]M.H.Protter and H.F.Weinberger, Maximum Principles in Differential Equations, Springer-verlag, NewYork,1994
[15]T.Aubin, "Nonlinear Analysis on Manifolds, Monge-Ampere Equations," Springer-Verlag, New York,1982.
[16]R.S.Hamilton, Three-manifolds with positive Ricci curvature, J.Differential Geom, 1982,17:255-306.
[17]V.I.Oliker, Evolution of nonparametric surfaces with speed depending On curva-ture,I.The Gauss curvature case, Indiana Univ.Math.J.,1991,40:237-258.
[18]]S.-T.Yau, On the Ricci curvature of compact kahler Manifold and the complex Monge-Ampere equations,I, Comm.Pure Appl.Math.,1978,31:339-411.
[19]F.Schulz, "Regularity Theory for Quasilinear Elliptic Systems and Monge-Ampere Equations in Two Dimensions", Springer-verlag, Berlin/Heidlberg,1990.
[20]Y.Y.Li, Some existence results of fully nonlinear elliptic equation of Monge-Ampere type, Comm.Pure Applied Math.,1990,43:233-271.
[21]N.S.Trudinger, On the Dirichlet problem for hessian equation, preprint.
[22]N.S.Trudinger and J.I.E.Urbas, On second derivative estimates for equations of Monge-Ampere type, Bull.Austral.Math.,1984,30:321-332.
[23]L.Ca.,L.Nr.and J.Spruck,The Dirichlet problem for nonlinear seconder-order elliptic equations Ⅰ:Monge-Ampere Equations, Comm.Pure Appl.Math.,1984,37:339-402.
[24]L.Ca.,L.Nr.and J.Spruck, The Dirichlet problem for nonlinear seconder-order elliptic equations Ⅱ:Complex Monge-Ampere and uniformly elliptic equations, Comm.Pure Appl.Math.,1985,38:209-252.
[25]L.Ca.,L.Nr.and J.Spruck, The Dirichlet problem for nonlinear seconder-order elliptic equations Ⅲ:Functions of the eigenvalues of the Hessian, Acta.Math.,1985,155:61-301.
[26]L.Ca.,L.Ni.,and J.Spruck, Nonlinear second-order elliptic equations V:The Dirichlet problem for Weingarten hypersurfaces, Comm.Pure Appl.Math.,1988,41:47-70.
[27]E.Cababi, Improper affine hyperspheres of convex type and a generalization of a theo-rem by K. Jorgens, Michigan Math.J.,1958,5:105-126.
[28]S.Y.Cheng and S.T.Yau, On the regularity of the solutions of the n-dimensional Minkowski problem, Comm.Pure Appl.Math.,1976,29:495-516.
[29]P.Delanoe, Equations du type Monge-Ampere sur les Varietes Riemanniennes com-pactes Ⅰ, J.Funct.Anal.,1980,40:358-386.
[30]P.Delanoe, Equations du type Monge-Ampere sur les Varietes Riemanniennes com-pactes Ⅱ, J.Funct.Anal.,1981,41:341-353.
[31]P.Delanoe, Equations du type Monge-Ampere sur les Varietes Riemanniennes com- pactes Ⅲ, J.Funct.Anal.,1982,45:403-430.
[32]S.D.Eidels'man, Parabolic Systems, North-Holland, Amsterdam,1969.
[33]Wang, G.L., The first boundary value problem for parabolic Monge-Ampere equation, Northeast Math.J.,1987,3(4):463-478.
[34]Wang, R.H.and Wang, G.L., On existence,uniqueness and regularity of viscosity so-lutions for the first boundary value problem to parabolic Monge-Ampere equation, Northeast.Math.,1992,8(4):417-446.
[35]D.Gilbarg and N.S. Trudinger, Elloptic partial differential equations of second-order, Springer-Verlag, New York,1983.
[36]N.S.Trudinger, The Dirichlet problem for the prescribed curvature equations, Arch.Rational Mech.Anal,1990,111:153-179.
[37]N.V.Krylov, Nolinear Elliptic and Parabolic Equations of the Second Ordre, Reidel, Boston, Ma.,1984.
[38]Guan.B.and Spruck,J., Boundary value problem on Sn for surfaces of constant Guass curvature, Ann.of Math.,1993,138:601-624.
[39]陈维桓,微分流形初步,高等教育出版社,2004.
[40]陈维桓,李兴校,黎曼几何引论(上),北京大学出版社,2002.
[41]陈省身,陈维桓,微分几何讲义,高等教育出版社,2002.
[42]李开泰,黄艾香,张量分析及其运用,西安交通大学出版社,2006.
[43]白正国,沈一兵,水乃翔,郭孝英,黎曼几何初步,高等教育出版社,2004.
[44]张恭庆,林源渠,泛函分析讲义(上册),北京,北京大学出版社,1987.
[45]伍卓群,尹景学,王春明,椭圆与抛物型方程引论,科学出版社,2002.
[46]陈恕行,现代偏微分方程引论,科学出版社,2005.
[47]叶其孝,李正元,反应扩散方程,科学出版社,1990.
[2]Ahmed Mohammed,Existence and estimates of solutions to a singular Dirichlet problem for the Monge-Ampere equation, J.Math.Anal.Appl.,2008,340:1226-1234.
[3]H.-D. Cao, Deformation of Kahler metrics to Kahlealer-Einstein metrics on compact kahler manifolds, Invent.Math.,1958,81:359-372.
[4]Barbara Huisken, Parabolic Monge-Ampere Equations on Riemanian Manifolds, Jour-nal of Functional Analysis,1997,147:140-163.
[5]Bo Guan and Yan Yan Li, Monge-Ampere Equations on Riemannian Manifolds, Journal Of Differential Equations,1996,132:126-139.
[6]Lian Songzhe, Existence of solution to initial value problem for a parabolic Monge-Ampere equation and application, Nonlinear Analysis,2006,65:59-78.
[7]L.Caffarelli,YanYan Li, A Liouville theorem for solution of the Monge-Ampere equation with periodic data, Ann.I.H.Poin.,2004,21:97-120.
[8]Ni Xiang,Xiao-Ping Yang, The complex Monge-Ampere equation with infinite bound-ary value, Nonlinear Analysis,2008,68:1075-1081.
[9]I.J.Bakelman, Convex Analysis on Manifolds, Monge-Ampere Equations, Springer-Verlag, Berlin/Heidelberg,1980.
[10]Qingbo Huang, On the mean oscillation of the Hessian of solutions to the Monge-Ampere equation, Advances in Mathematics,2006,207:599-616.
[11]R.Chang yu and W.Guang lie, Parabolic Monge-Ampere Equation on Riemannian Manifolds, Northeast Math.J.,2007,23(5):413-424.
[12]N.S.Trudinger and X.J.Wang, Boundary regularity for the Monge-Ampere and affine maximal surface equations, Ann.of Math.,2008,167:993-1028.
[13]A.D.Alexandrov, Dirichlets'problem for the equation Det‖Zij‖=φ, Vestnik Leningrad Univ.,1985,13, No.1:5-24.[In Russian]
[14]M.H.Protter and H.F.Weinberger, Maximum Principles in Differential Equations, Springer-verlag, NewYork,1994
[15]T.Aubin, "Nonlinear Analysis on Manifolds, Monge-Ampere Equations," Springer-Verlag, New York,1982.
[16]R.S.Hamilton, Three-manifolds with positive Ricci curvature, J.Differential Geom, 1982,17:255-306.
[17]V.I.Oliker, Evolution of nonparametric surfaces with speed depending On curva-ture,I.The Gauss curvature case, Indiana Univ.Math.J.,1991,40:237-258.
[18]]S.-T.Yau, On the Ricci curvature of compact kahler Manifold and the complex Monge-Ampere equations,I, Comm.Pure Appl.Math.,1978,31:339-411.
[19]F.Schulz, "Regularity Theory for Quasilinear Elliptic Systems and Monge-Ampere Equations in Two Dimensions", Springer-verlag, Berlin/Heidlberg,1990.
[20]Y.Y.Li, Some existence results of fully nonlinear elliptic equation of Monge-Ampere type, Comm.Pure Applied Math.,1990,43:233-271.
[21]N.S.Trudinger, On the Dirichlet problem for hessian equation, preprint.
[22]N.S.Trudinger and J.I.E.Urbas, On second derivative estimates for equations of Monge-Ampere type, Bull.Austral.Math.,1984,30:321-332.
[23]L.Ca.,L.Nr.and J.Spruck,The Dirichlet problem for nonlinear seconder-order elliptic equations Ⅰ:Monge-Ampere Equations, Comm.Pure Appl.Math.,1984,37:339-402.
[24]L.Ca.,L.Nr.and J.Spruck, The Dirichlet problem for nonlinear seconder-order elliptic equations Ⅱ:Complex Monge-Ampere and uniformly elliptic equations, Comm.Pure Appl.Math.,1985,38:209-252.
[25]L.Ca.,L.Nr.and J.Spruck, The Dirichlet problem for nonlinear seconder-order elliptic equations Ⅲ:Functions of the eigenvalues of the Hessian, Acta.Math.,1985,155:61-301.
[26]L.Ca.,L.Ni.,and J.Spruck, Nonlinear second-order elliptic equations V:The Dirichlet problem for Weingarten hypersurfaces, Comm.Pure Appl.Math.,1988,41:47-70.
[27]E.Cababi, Improper affine hyperspheres of convex type and a generalization of a theo-rem by K. Jorgens, Michigan Math.J.,1958,5:105-126.
[28]S.Y.Cheng and S.T.Yau, On the regularity of the solutions of the n-dimensional Minkowski problem, Comm.Pure Appl.Math.,1976,29:495-516.
[29]P.Delanoe, Equations du type Monge-Ampere sur les Varietes Riemanniennes com-pactes Ⅰ, J.Funct.Anal.,1980,40:358-386.
[30]P.Delanoe, Equations du type Monge-Ampere sur les Varietes Riemanniennes com-pactes Ⅱ, J.Funct.Anal.,1981,41:341-353.
[31]P.Delanoe, Equations du type Monge-Ampere sur les Varietes Riemanniennes com- pactes Ⅲ, J.Funct.Anal.,1982,45:403-430.
[32]S.D.Eidels'man, Parabolic Systems, North-Holland, Amsterdam,1969.
[33]Wang, G.L., The first boundary value problem for parabolic Monge-Ampere equation, Northeast Math.J.,1987,3(4):463-478.
[34]Wang, R.H.and Wang, G.L., On existence,uniqueness and regularity of viscosity so-lutions for the first boundary value problem to parabolic Monge-Ampere equation, Northeast.Math.,1992,8(4):417-446.
[35]D.Gilbarg and N.S. Trudinger, Elloptic partial differential equations of second-order, Springer-Verlag, New York,1983.
[36]N.S.Trudinger, The Dirichlet problem for the prescribed curvature equations, Arch.Rational Mech.Anal,1990,111:153-179.
[37]N.V.Krylov, Nolinear Elliptic and Parabolic Equations of the Second Ordre, Reidel, Boston, Ma.,1984.
[38]Guan.B.and Spruck,J., Boundary value problem on Sn for surfaces of constant Guass curvature, Ann.of Math.,1993,138:601-624.
[39]陈维桓,微分流形初步,高等教育出版社,2004.
[40]陈维桓,李兴校,黎曼几何引论(上),北京大学出版社,2002.
[41]陈省身,陈维桓,微分几何讲义,高等教育出版社,2002.
[42]李开泰,黄艾香,张量分析及其运用,西安交通大学出版社,2006.
[43]白正国,沈一兵,水乃翔,郭孝英,黎曼几何初步,高等教育出版社,2004.
[44]张恭庆,林源渠,泛函分析讲义(上册),北京,北京大学出版社,1987.
[45]伍卓群,尹景学,王春明,椭圆与抛物型方程引论,科学出版社,2002.
[46]陈恕行,现代偏微分方程引论,科学出版社,2005.
[47]叶其孝,李正元,反应扩散方程,科学出版社,1990.