A “quasi maximum principle” for $I$-surfaces

Abstract

The result of this paper yields a maximum principle for the components of surfaces whose distortion by a certain ${\mathrm{GL}}_3(\mathbb{R})$ matrix are minimizers of a dominance functional $\mathcal{I}$ of a parametric functional $\mathcal{J}$ with dominant area term within boundary value classes $H_{\varphi }^{1,2}(B,{\mathbb{R}}^3)$, termed $\mathcal{I}$-surfaces. Finally we derive a compactness result for sequences of $\mathcal{I}$-surfaces in $C^0(B,{\mathbb{R}}^3)$, which serves as a preparation for the forthcoming article [R. Jakob, Unstable extremal surfaces of the “Shiffman functional” spanning rectifiable boundary curves, Calc. Var., submitted for publication] whose aim is a proof of a sufficient condition for the existence of extremal surfaces of $\mathcal{J}$ which do not furnish global minima of $\mathcal{J}$ within the class ${\mathcal{C}}^{\ast }(\Gamma )$ of $H^{1,2}$-surfaces spanning an arbitrary closed rectifiable boundary curve $\Gamma \subset {\mathbb{R}}^3$ that merely has to satisfy a chord-arc condition.