Minimizing Movements for the Generalized Power Mean Curvature Flow

Area of Science:

• Geometric Analysis
• Partial Differential Equations
• Calculus of Variations

Background:

• The Almgren-Taylor-Wang scheme is a method for mean curvature flow.
• De Giorgi's conjecture motivates exploring generalized penalization terms.
• Understanding the convergence of discrete schemes to continuous evolution equations is crucial.

Purpose of the Study:

• To analyze a generalized minimizing movement scheme for mean curvature flow.
• To investigate the convergence of this scheme to a specific geometric evolution equation.
• To extend the analysis to anisotropic settings and driving forces.

Main Methods:

• Utilizing a generalized minimizing movement scheme with a novel integral penalization term.
• Analyzing the convergence properties of the scheme.
• Extending the framework to anisotropic mean curvature flow and flows with driving forces.

Main Results:

• The generalized minimizing movement scheme converges to the geometric evolution equation f(v) = -κ on ∂E(t).
• The analysis is extended to anisotropic settings and flows with driving forces.
• Minimizing movements are shown to coincide with smooth classical solutions when they exist.
• Mean convexity and convexity are preserved by the weak flow in the absence of forcing.

Conclusions:

• The proposed generalized minimizing movement scheme provides a robust framework for studying mean curvature flow.
• The convergence to the geometric evolution equation is established for a broad class of functions f.
• The study offers insights into the behavior of geometric evolution equations in various settings.