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Flat minimizers of the Willmore functional: Euler-Lagrange equations
Let $S\subset R^2$ be a bounded $C^1$ domain and let $g$ denote the flat metric in $R^2$.
We prove that there exist minimizers of the Willmore functional restricted to a class of isometric immersions of the Riemannian surface $(S, g)$ into $R^3$.
We derive the Euler-Lagrange equations satisfied by such constrained minimizers. Our main motivation comes from nonlinear elasticity, where this constrained Willmore functional arises naturally and is called Kirchhoff's plate functional.