On the second boundary value problem for a class of fully nonlinear flows II

Abstract

This article is a continuation of earlier work [R.L. Huang and Y.H. Ye, On the second boundary value problem for a class of fully nonlinear flows I, to appear in International Mathematics Research Notices], where the long time existence and convergence were given on some general parabolic type special Lagrangian equations. The long time existence and convergence of the flow had been obtained in all cases. In particular, we can prescribe the second boundary value problems for a family of special Lagrangian graphs.(1.1) F τ (D 2 u) = c, x ∈ Ω,

On the second boundary value problem for a class of fully nonlinear flows III

“…For τ ∈ 0, π 2 , if |κ| is sufficiently small, then there exist a uniformly convex solution u ∈ C ∞ ( Ω) and a unique constant c solving (1.4), and u is unique up to a constant. Theorem 1.1 exhibits an extension of the previous work on κ = 0 done by Brendle-Warren [10], Huang [11], Huang-Ou [12], Huang-Ye [13] and Chen-Huang-Ye [14].…”

“…As far as τ = π 2 is concerned, Brendle and Warren [10] proved the existence and uniqueness of the solution by the elliptic method, and the second author [11] obtained the existence of solution by considering the second boundary value problem for Lagrangian mean curvature flow. Then by the elliptic and parabolic method, the second author with Ou [12], Ye [13] and Chen [14] proved the existence and uniqueness of the solution for 0 < τ < π 2 . We are now in a position to find out the Lagrangian graph (x, Du(x)) prescribed constant mean curvature vector κ in (R n × R n , g τ ) such that Du is the diffeomorphism between two uniformly convex bounded domains.…”

On the second boundary value problem for Lagrangian mean curvature equation

“…Theorem 1.3 presents an extension of the previous work on κ = 0 done by Brendle-Warren [9], Huang [10], Huang-Ou [11], Huang-Ye [12] and Chen-Huang-Ye [13].…”

“…As far as τ = π 2 is concerned, Brendle and Warren [9] proved the existence and uniqueness of the solution by the elliptic method, and the second author [10] obtained the existence of solution by the parabolic method. Then by the elliptic and parabolic method, the second author with Ou [11], Ye [12] [13] and Chen [13] proved the existence and uniqueness of the solution to (1.7) for 0 < τ < π 2 . For a smooth function f , Chen, Zhang and the third author [14] proved that if u satisfies…”

On the second boundary value problem for special Lagrangian curvature potential equation

“…Using the parabolic method, Schnürer and Smoczyk [11] also arrived at the existence of solutions to (1.1) for τ = 0. For further studies on the rest of 0 ≤ τ ≤ π 2 , see [1], [12] [13] and the references therein.…”