外区域上的抛物型Monge-Ampère方程
|
摘要
研究外区域上的抛物型Monge-Ampere方程-u_t det(D~2u)=f解的存在性.利用Perron方法得到了该方程的外问题具有渐近性质解的存在性与唯一性.
We study the existence of solutions to the parabolic Monge-Ampere equations —u_t det(D~2u) = / on the exterior domains.Using the Perron method,we get the existence and uniqueness of solutions with the asymptotic behavior to the exterior problems of parabolic Monge-Ampere equations.
We study the existence of solutions to the parabolic Monge-Ampere equations —u_t det(D~2u) = / on the exterior domains.Using the Perron method,we get the existence and uniqueness of solutions with the asymptotic behavior to the exterior problems of parabolic Monge-Ampere equations.
引文
[1]Bao J.G.,Li H.G.,Li Y.Y.,On the exterior Dirichlet problem for Hessian equations,Trans.Amer.Math.Soc,2014,366:6183-6200.
[2]Bao J.G.,Li H.G.,Zhang L.,Monge-Ampere equation on exterior domains,Cole.Var.Partial Differential Equations,2015,52:39-63.
[3]Caffarelli L.A.,Interior W2'v estimates for solutions of the Monge-Ampere equation,Ann.of Math.,1990,131:135-150.
[4]Caffarelli L.,Li Y.Y.,An extension to a theorem of Jorgens,Calabi,and Pogorelov,Comm.Pure Appl.Math.,2003,56:549-583.
[5]Calabi E.,Improper affine hyperspheres of convex type and a generalization of a theorem by K.Jorgens,Michigan Math.J.,1958,5:105-126.
[6]Cheng S.Y.,Yau S.T.,Complete affine hypersurfaces,I.The completeness of affine metrics,Comm.Pure Appl.Math.,1986,39:839-866.
[7]Dai L.M.,Exterior problems to fully nonlinear uniformly elliptic equations,Chinese Ann.Math.Ser.A,2011,32A(6):721-728(in Chinese).
[8]Dai L.M.,Bao J.G.,On uniqueness and existence of viscosity solutions to Hessian equations in exterior domains,Front.Math.China,2011,6:221-230.
[9]Gutierrez C.E.,Huang Q.B.,A generalization of a theorem by Calabi to the parabolic Monge-Ampere equation,Indiana Univ.Math.J.,1998,47:1459-1480.
[10]Jorgens K.,Uber die Losungen der Differentialgleichung rt-s~2=1(German),Math.Ann.,1954,127:130-134.
[11]Ju H.J.,Bao J.G.,On the exterior Dirichlet problem for Monge-Ampere equations,J.Math.Anal.Appl.,2013,405:475-483.
[12]Krylov N.V.,Sequences of convex functions and estimates of the maximum of the solution of a parabolic equation(Russian),Sibirsk.Mat.Zh.,1976,17:290-303.
[13]Krylov N.V.,On controllable diffusion processes with unbounded coefficients(Russian),Izv.Akad.Nauk SSSR Ser.Mat,1981,45:734-759.
[14]Krylov N.V.,On the control of a diffusion process until the moment of the first exit from the domain(Russian),Izv.Akad.Nauk SSSR Ser.Mat,1981,45:1029-1048.
[15]Krylov N.V.,Boundedly inhomogeneous elliptic and parabolic equations(Russian),Izv.Akad.Nauk SSSR Ser.Mat,1982,46:487-523.
[16]Li H.G.,Bao J.G.,On the exterior Dirichlet problem for the Monge-Ampere equation in dimension two,Nonlinear Anal,2012,75:6448-6455.
[17]Li H.G.,Bao J.G.,The exterior Dirichlet problem for special Lagrangian equations in dimensions n<4,Nonlinear Anal.,2013,89:219-229.
[18]Li H.G.,Bao J.G.,The exterior Dirichlet problem for fully nonlinear elliptic equations related to the eigenvalues of the Hessian,J.Differential Equations,2014,256:2480-2501.
[19]Li H.G.,Dai L.M.,The exterior Dirichlet problem for Hessian quotient equations,J.Math.Anal.Appl.,2012,393:534-543.
[20]Pogorelov A.V.,On the improper convex affine hyperspheres,Geometriae Dedicata,1972,1:33-46.
[21]Tso K.,On a real Monge-Ampere functional,Invent.Math.,1990,101:425-448.
[22]Wang G.L.,The first boundary value problem for parabolic Monge-Ampere equation,Northeast.Math.J.,1987,3:463-478.
[23]Wang R.H.,Wang G.L.,On existence,uniqueness and regularity of viscosity solutions for the first initialboundary value problems to parabolic Monge-Ampere equation,Northeast.Math.J.,1992,8:417-446.
[24]Wang R.H.,Wang G.L.,The geometric measure theoretical characterization of viscosity solutions to parabolic Monge-Ampere type equation,J.Partial Differential Equations,1993,6:237-254.
[25]Xiong J.G.,Bao J.G.,On Jorgens,Calabi,and Pogorelov type theorem and isolated singularities of parabolic Monge-Ampere equations,J.Differential Equations,2011,250:367-385.
[2]Bao J.G.,Li H.G.,Zhang L.,Monge-Ampere equation on exterior domains,Cole.Var.Partial Differential Equations,2015,52:39-63.
[3]Caffarelli L.A.,Interior W2'v estimates for solutions of the Monge-Ampere equation,Ann.of Math.,1990,131:135-150.
[4]Caffarelli L.,Li Y.Y.,An extension to a theorem of Jorgens,Calabi,and Pogorelov,Comm.Pure Appl.Math.,2003,56:549-583.
[5]Calabi E.,Improper affine hyperspheres of convex type and a generalization of a theorem by K.Jorgens,Michigan Math.J.,1958,5:105-126.
[6]Cheng S.Y.,Yau S.T.,Complete affine hypersurfaces,I.The completeness of affine metrics,Comm.Pure Appl.Math.,1986,39:839-866.
[7]Dai L.M.,Exterior problems to fully nonlinear uniformly elliptic equations,Chinese Ann.Math.Ser.A,2011,32A(6):721-728(in Chinese).
[8]Dai L.M.,Bao J.G.,On uniqueness and existence of viscosity solutions to Hessian equations in exterior domains,Front.Math.China,2011,6:221-230.
[9]Gutierrez C.E.,Huang Q.B.,A generalization of a theorem by Calabi to the parabolic Monge-Ampere equation,Indiana Univ.Math.J.,1998,47:1459-1480.
[10]Jorgens K.,Uber die Losungen der Differentialgleichung rt-s~2=1(German),Math.Ann.,1954,127:130-134.
[11]Ju H.J.,Bao J.G.,On the exterior Dirichlet problem for Monge-Ampere equations,J.Math.Anal.Appl.,2013,405:475-483.
[12]Krylov N.V.,Sequences of convex functions and estimates of the maximum of the solution of a parabolic equation(Russian),Sibirsk.Mat.Zh.,1976,17:290-303.
[13]Krylov N.V.,On controllable diffusion processes with unbounded coefficients(Russian),Izv.Akad.Nauk SSSR Ser.Mat,1981,45:734-759.
[14]Krylov N.V.,On the control of a diffusion process until the moment of the first exit from the domain(Russian),Izv.Akad.Nauk SSSR Ser.Mat,1981,45:1029-1048.
[15]Krylov N.V.,Boundedly inhomogeneous elliptic and parabolic equations(Russian),Izv.Akad.Nauk SSSR Ser.Mat,1982,46:487-523.
[16]Li H.G.,Bao J.G.,On the exterior Dirichlet problem for the Monge-Ampere equation in dimension two,Nonlinear Anal,2012,75:6448-6455.
[17]Li H.G.,Bao J.G.,The exterior Dirichlet problem for special Lagrangian equations in dimensions n<4,Nonlinear Anal.,2013,89:219-229.
[18]Li H.G.,Bao J.G.,The exterior Dirichlet problem for fully nonlinear elliptic equations related to the eigenvalues of the Hessian,J.Differential Equations,2014,256:2480-2501.
[19]Li H.G.,Dai L.M.,The exterior Dirichlet problem for Hessian quotient equations,J.Math.Anal.Appl.,2012,393:534-543.
[20]Pogorelov A.V.,On the improper convex affine hyperspheres,Geometriae Dedicata,1972,1:33-46.
[21]Tso K.,On a real Monge-Ampere functional,Invent.Math.,1990,101:425-448.
[22]Wang G.L.,The first boundary value problem for parabolic Monge-Ampere equation,Northeast.Math.J.,1987,3:463-478.
[23]Wang R.H.,Wang G.L.,On existence,uniqueness and regularity of viscosity solutions for the first initialboundary value problems to parabolic Monge-Ampere equation,Northeast.Math.J.,1992,8:417-446.
[24]Wang R.H.,Wang G.L.,The geometric measure theoretical characterization of viscosity solutions to parabolic Monge-Ampere type equation,J.Partial Differential Equations,1993,6:237-254.
[25]Xiong J.G.,Bao J.G.,On Jorgens,Calabi,and Pogorelov type theorem and isolated singularities of parabolic Monge-Ampere equations,J.Differential Equations,2011,250:367-385.