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

TL;DR

This paper studies a class of fully nonlinear parabolic flows with nonlinear Neumann boundary conditions, proving convexity preservation, long-term existence, and convergence, and enabling the prescription of second boundary value problems for special Lagrangian graphs.

Contribution

It extends the analysis of fully nonlinear flows to include nonlinear boundary conditions, establishing convexity preservation and long-term behavior for these flows.

Findings

Convexity is preserved for solutions of the flows.

The flows exist for all time and converge.

Second boundary value problems can be prescribed for special Lagrangian graphs.

Abstract

In this paper, a class of fully nonlinear flows with nonlinear Neumann type boundary condition is considered. This problem was solved partly by the first author under the assumption that the flow is the parabolic type special Lagrangian equation in $\mathbb{R}^{2n}$. We show that the convexity is preserved for solutions of the fully nonlinear parabolic equations and prove the long time existence and convergence of the flow. In particular, we can prescribe the second boundary value problems for a family of special Lagrangian graphs in Euclidean and pseudo-Euclidean space.