Confined Willmore energy and the area functional (Q6185122)

The author studies the minimization problem: \((P)_{\Omega ,\Lambda }:\min\{ \mathcal{W}_{\Lambda }(\Sigma ):\Sigma \subset \overline{\Omega }\}\), where \( \Omega \subset \mathbb{R}^{3}\) is open, bounded and with \(C^{2}\)-boundary \( \partial \Omega \), \(\Sigma \subset \mathbb{R}^{3}\) is a smooth surface, and \( \mathcal{W}_{\Lambda }\) is the functional defined as: \(\mathcal{W}_{\Lambda }(\Sigma )=\mathcal{W}(\Sigma )-\Lambda \left\vert \Sigma \right\vert \), \( \Lambda >0\) being fixed, \(\mathcal{W}(\Sigma )\) denoting the Willmore energy, and \(\left\vert \Sigma \right\vert \) denoting the area of \(\Sigma \). The author takes \(\Omega \subset B_{1/2}(0)\) (so that \( \mathrm{diam}(\Omega )\leq 1\)). He lets \(C_{\Lambda }=\mathrm{inf}\{\mathcal{W}(\Sigma ):\Sigma \subset \overline{\Omega }\}\), which defines a function \(C_{\Lambda }\) from \((0,+\infty )\) to \([-\infty ,+\infty )\), further he denotes \(\Lambda _{\Omega }=\mathrm{inf}\{\Lambda >0:C_{\Lambda }=-\infty \}\), \(\mathbb{W}(\Sigma )=\mathcal{W} (\Sigma )/\left\vert \Sigma \right\vert \), and \(\overline{\Lambda }_{\Omega }=\mathrm{inf}\{\mathbb{W}(\Sigma ):\Sigma \subset \overline{\Omega }\}\). The main result proves that \(C_{\Lambda }\) is a concave, continuous, non-negative, and strictly decreasing function on an interval \((0,\Lambda _{\Omega }]\) for some \(\Lambda _{\Omega }\in \lbrack 4,1/\varepsilon _{\Omega }^{2}]\), where \( \Lambda _{\Omega }\) and \(\varepsilon _{\Omega }\) depend on \(\Omega \). Moreover \(\lim_{\Lambda \rightarrow 0^{+}}C_{\Lambda }=4\pi \), \(C_{\Lambda _{\Omega }}>0\) and \(C_{\Lambda }=-\infty \) for all \(\Lambda >\Lambda _{\Omega }\). If \(\Lambda \in (0,\Lambda _{\Omega })\), there exists a minimizing sequence \((\Sigma _{n}^{\Lambda })_{n}\) for the functional \( \mathcal{W}_{\Lambda }\) which converges in the sense of varifolds to a varifold \(V\) that is integer rectifiable and whose generalized mean curvature is square integrable with respect to the weight measure of \(V\). If \(\Lambda \) is sufficiently small, depending only on \(\Omega \), the limit varifold previously indicated is a \(C^{1,\alpha }\cap W^{2,2}\) embedded surface with multiplicity one and is such that \(\mathcal{W}_{\Lambda }(\Sigma )=C_{\Lambda }\). Moreover \(\Sigma \cap \Omega \) is of class \(C^{\infty }\). This general result is specialized in the case where \(\Omega =B_{1}\), the unit ball of \(\mathbb{R}^{3}\). For the proof, the author first recalls properties of the Willmore functional \(\mathcal{W}(\cdot )\): lower semicontinuity, conformal invariance, existence of a lower bound for immersed surfaces, then bounds between this Willmore functional, and properties of the \(\mathcal{W}_{\Lambda }\)-energy. Throughout the proof, the author presents examples.