The first initial–boundary value problem for Hessian equations of parabolic type on Riemannian manifolds

Abstract

MSC:Keywords: Riemannian manifolds Fully nonlinear parabolic equations First initial-boundary value problem a priori estimates a b s t r a c tIn this paper, we are concerned with the first initial-boundary value problem for a class of fully nonlinear parabolic equations on Riemannian manifolds. As usual, the establishment of the a priori C 2 estimates is our main part. Based on these estimates, the existence of classical solutions is proved under conditions which are nearly optimal.

Second order estimates for Hessian equations of parabolic type on Riemannian manifolds

“…Define M T = M × (0, T ] ⊂ M × R, PM T = BM T ∪ SM T is the parabolic boundary of M T with BM T = M × {0} and SM T = ∂M × [0, T ]. In [3], the authors derived C 2 estimates for solutions of the first initial-boundary value problem of parabolic Hessian equations in the form (1.1) f (λ(∇ 2 u + χ(x, t)), −u t ) = ψ(x, t),…”