Boundary behavior of solutions of the mean curvature flow with boundary conditions and nonlinear degenerate parabolic equations

2015 Fiscal Year Final Research Report

• Project/Area Number: 25800084
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Mathematical analysis
• Research Institution: Nihon University
• Principal Investigator: Mizuno Masashi 日本大学, 理工学部, 准教授 (80609545)
• Project Period (FY): 2013-04-01 – 2016-03-31
• Keywords: 境界挙動 / 平均曲率流方程式 / 幾何学的測度論 / 特異極限問題

Outline of Final Research Achievements

We study the singular limit problem for the Allen-Cahn equation with Neumann boundary conditions and prove that associated energy measure converges to a measure theoretic solution of the mean curvature flow with right-angle boundary conditions. In particular, we establish the right-angle boundary condition in geometric measure theory and prove the boundary monotonicity formula for the energy measure.

Next, we study the mean curvature flow of graphs both with Neumann boundary conditions and transport terms. We derive a priori boundary gradient estimate for solutions and then show time local existence of solutions of the mean curvature flow under the natural assumption from the point of scaling arguments for the transport terms.

Free Research Field

偏微分方程式論