On a $\Gamma$-limit of Willmore functionals with additional curvature penalization term

TL;DR

This paper studies a modified Willmore functional with a curvature-based penalization term on graphs, analyzing its limit behavior as the penalization becomes large through $ ext{Gamma}$-convergence.

Contribution

It introduces a new constrained version of the Willmore functional and derives its limit functional via $ ext{Gamma}$-convergence as penalization increases.

Findings

Derived the $ ext{Gamma}$-limit of the penalized Willmore functional.

Established the connection between penalization parameter and constraint enforcement.

Provided a rigorous mathematical framework for the limit behavior of curvature penalized functionals.

Abstract

We consider the Willmore functional on graphs, with an additional penalization of the area where the curvature is non-zero. Interpreting the penalization parameter as a Lagrange multiplier, this corresponds to the Willmore functional with a constraint on the area where the graph is flat. Sending the penalization parameter to $\infty$ and rescaling suitably, we derive the limit functional in the sense of $\Gamma$-convergence.