On the constancy theorem for anisotropic energies through differential inclusions

Area of Science:

• Geometric measure theory
• Differential geometry
• Calculus of variations

Background:

• Stationary graphs are crucial in geometric analysis, particularly for understanding minimal surfaces and related structures.
• The concept of currents and varifolds provides a framework for studying geometric objects with lower regularity.
• Differential inclusions offer a novel approach to characterizing stationarity conditions.

Purpose of the Study:

• To reformulate the stationarity condition for graphs with multiplicity as a differential inclusion.
• To analyze the existence of specific configurations within this framework under different hypotheses on the integrand.
• To investigate the construction of degenerate stationary points.

Main Methods:

• Utilizing the framework of differential inclusions applied to geometric functionals.
• Defining a set of matrices to translate stationarity into a differential inclusion.
• Applying convex integration techniques to construct degenerate solutions.

Main Results:

• Proving that for non-negative integrands, no degenerate configurations exist, corroborating prior work.
• Demonstrating the existence of degenerate configurations when the non-negativity hypothesis is removed.
• Constructing a highly degenerate stationary point with multiplicity via convex integration.

Conclusions:

• The differential inclusion approach effectively characterizes stationarity for graphs.
• The non-negativity of the integrand plays a critical role in the existence of degenerate configurations.
• Convex integration provides a powerful tool for constructing complex geometric structures.