The Neumann Problem for Parabolic Hessian Quotient Equations

In this paper, we consider the Neumann problem for parabolic Hessian quotient equations. We show that the $k$-admissible solution of the parabolic Hessian quotient equation exists for all time and converges to the smooth solution of elliptic Hessian quotient equations. Also the solutions of the classical Neumann problem converge to a translating solution.

[1]  Xinan Ma,et al.  Gradient Estimates of Mean Curvature Equations with Neumann Boundary Condition , 2014, 1406.0046.

[2]  H. Schwetlick,et al.  Translating solutions for Gaußcurvature flows with Neumann boundary conditions , 2004 .

[3]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[4]  Dekai Zhang,et al.  The Neumann problem of Hessian quotient equations , 2020, 2003.10875.

[5]  G. Huisken,et al.  Convexity estimates for mean curvature flow and singularities of mean convex surfaces , 1999 .

[6]  Yanyan Li,et al.  A fully nonlinear version of the Yamabe problem on locally conformally flat manifolds with umbilic boundary , 2009, 0911.3366.

[7]  Yanyan Li.  Interior gradient estimates for solutions of certain fully nonlinear elliptic equations , 1991 .

[8]  J. Spruck,et al.  The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian , 1985 .

[9]  Oliver C. Schnuerer,et al.  Translating solutions for Gauss curvature flows with Neumann boundary conditions , 2003 .

[10]  G. M. Lieberman.  SECOND ORDER PARABOLIC DIFFERENTIAL EQUATIONS , 1996 .

[11]  J. Spruck,et al.  Existence and regularity of a capillary surface with prescribed contact angle , 1976 .

[12]  Bo Guan,et al.  SECOND-ORDER ESTIMATES AND REGULARITY FOR FULLY NONLINEAR ELLIPTIC EQUATIONS ON RIEMANNIAN MANIFOLDS , 2012, 1211.0181.

[13]  Carlos E. Kenig,et al.  Boundary value problems for elliptic equations , 1991 .

[14]  J. Spruck,et al.  Nonlinear Second-Order Elliptic Equations V. The Dirichlet Problem for Weingarten Hypersurfaces , 1988 .

[15]  Neil S. Trudinger,et al.  On the Dirichlet problem for Hessian equations , 1995 .

[16]  José F. Escobar,et al.  The Yamabe problem on manifolds with boundary , 1992 .

[17]  Luis A. Caffarelli,et al.  The Dirichlet problem for nonlinear second-order elliptic equations I , 1984 .

[18]  J. Spruck.  GEOMETRIC ASPECTS OF THE THEORY OF FULLY NON LINEAR ELLIPTIC EQUATIONS , 2003 .

[19]  Matthew J. Gursky,et al.  An equation of Monge-Ampère type in conformal geometry, and four-manifolds of positive Ricci curvature , 2002 .

[20]  Damin Wu,et al.  A second order estimate for complex Hessian equations on a compact Kähler manifold , 2010 .

[21]  N. Trudinger.  On degenerate fully nonlinear elliptic equations in balls , 1987, Bulletin of the Australian Mathematical Society.

[22]  Xinan Ma,et al.  Gradient estimates of mean curvature equations with Neumann boundary value problems , 2016 .

[23]  N. Trudinger,et al.  Regularity of Potential Functions of the Optimal Transportation Problem , 2005 .

[24]  Yanyan Li.  Some existence results for fully nonlinear elliptic equations of monge‐ampère type , 1990 .

[25]  J. Urbas.  Hessian equations on compact Riemannian manifolds , 2002 .

[26]  Lawrence C. Evans,et al.  Classical solutions of fully nonlinear, convex, second‐order elliptic equations , 1982 .

[27]  S. Dinew,et al.  Liouville and Calabi-Yau type theorems for complex Hessian equations , 2012, 1203.3995.

[28]  A. Pogorelov.  The Minkowski multidimensional problem , 1978 .

[29]  Pengfei Guan,et al.  Convex hypersurfaces of prescribed curvature , 2002 .

[30]  Knut Smoczyk,et al.  Neumann and second boundary value problems for Hessian and Gauß curvature flows , 2003 .

[31]  N. Krylov,et al.  BOUNDEDLY NONHOMOGENEOUS ELLIPTIC AND PARABOLIC EQUATIONS IN A DOMAIN , 1984 .

[32]  J. Urbas.  On the second boundary value problem for equations of Monge-Ampère type. , 1997 .

[33]  N. Trudinger,et al.  Oblique boundary value problems for augmented Hessian equations I , 2015, Communications in Partial Differential Equations.

[34]  Gary M. Lieberman,et al.  Nonlinear oblique boundary value problems for nonlinear elliptic equations , 1986 .

[35]  Xu-jia Wang,et al.  A variational theory of the Hessian equation , 2001 .

[36]  Yanyan Li,et al.  Estimates and existence results for a fully nonlinear Yamabe problem on manifolds with boundary , 2006, math/0604258.

[37]  Chuanqiang Chen.  The interior gradient estimate of Hessian quotient equations , 2015 .

[38]  C. Xia,et al.  Classical Neumann Problems for Hessian equations and Alexandrov-Fenchel's inequalities , 2016, 1607.03868.

[39]  Pengfei Guan,et al.  The Christoffel-Minkowski problem I: Convexity of solutions of a Hessian equation , 2003 .

[40]  Claus Gerhardt.  Global regularity of the solutions to the capillarity problem , 1976 .

[41]  J. Urbas.  Nonlinear oblique boundary value problems for Hessian equations in two dimensions , 1995 .

[42]  Oblique boundary value problems for augmented Hessian equations II , 2017 .

[43]  G. M. Lieberman,et al.  Oblique Derivative Problems for Elliptic Equations , 2013 .

[44]  Guohuan Qiu,et al.  The Neumann Problem for Hessian Equations , 2015, Communications in Mathematical Physics.

[45]  Ya-Zhe Chen,et al.  Second Order Elliptic Equations and Elliptic Systems , 1998 .

[46]  L. Caffarelli.  Boundary regularity of maps with convex potentials – II , 1996 .

[47]  Xi-Nan Ma,et al.  Constant mean curvature surfaces and mean curvature flow with non-zero Neumann boundary conditions on strictly convex domains , 2018 .

[48]  Pierre-Louis Lions,et al.  THE Neumann problem for equations of Monge-Ampère type , 1986 .

[49]  Weimin Sheng,et al.  Convex Hypersurfaces of prescribed Weingarten curvatures , 2004 .

[50]  Szu-yu Sophie Chen.  Boundary value problems for some fully nonlinear elliptic equations , 2006, math/0604080.

[51]  O. Schn,et al.  problems for Hessian and Gauss curvature flows , 2002 .

[52]  Jeff A. Viaclovsky.  Conformal geometry, contact geometry, and the calculus of variations , 2000 .