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Parametric surfaces with prescribed mean curvature. (English) Zbl 1072.53001

The paper under review is a survey article based on a series of lectures given by one of the authors. It contains an overview on some old and new problems concerning two-dimensional parametric surfaces in \({\mathbf R}^3\) with prescribed mean curvature. The prescribed mean curvature problem is the following: Given a mapping \(H : \mathbb R^3 \to \mathbb R\), find a two-dimensional surface \(M\) in \(\mathbb R^3\) such that for all points \(p \in M\), the mean curvature at \(p\) of \(M\) equals \(H(p)\). In case \(H \equiv 0\), this problem reduces to the classical Plateau problem if we consider boundaries. Let \(u : \mathcal U \subset \mathbb R^2 \to M\) be a parametrization of a two-dimensional regular surface \(M\) in \(\mathbb R^3\). Then it is well-known that the mean curvature \(H\) is expressed in terms of the first and second fundamental forms of \(u\). Moreover if we assume \(u\) is conformal, or equivalently, \[ |u_x|^2 - |u_y|^2 = 0 = u_x \cdot u_y, \] then the mean curvature is given by \[ \Delta u = 2 H(u) u_x \wedge u_y \quad \text{on }{\mathcal U}, \tag{1} \] which called the \(H\)-equation.
In section 3, the authors have discussed the classical Plateau problem: Given a Jordan curve \(\gamma\) in \(\mathbb R^3\), find a surface \(M\) such that \(\partial M = \gamma\) and having zero mean curvature at all points. The authors listed several methods concerning this problem and especially gave a little more detailed explanation about the Douglas-Radó method. Considering a conformal parametrization, the classical Plateau problem can be reduced to the following form: Find \(u \in C^0({\overline D}, \mathbb R^3) \cap C^2(D, \mathbb R^3)\) such that \[ \begin{cases} \Delta u = 0 &\text{in } D\\ |u_x|^2 - |u_y|^2 = 0 = u_x \cdot u_y &\text{in }D\\ u_{|_{\partial D}}\text{ monotone parametrization of }\gamma, \end{cases}\tag \(P_0\) \] where \(D\) is a disk in \(\mathbb R^2\). Introducing the energy functional \(E_D(v) = \frac{1}{2} \int_D |\nabla v|^2\) and an appropriate function space, the solutions to \((P_0)\) can be found by solving a minimizing problem on \(E_0\).
In section 4, the authors have mentioned the Plateau problem for \(H\)-surfaces of nonzero \(H\) with a small solution as a generalization of the classical Plateau problem. Namely, given a Jordan curve \(\gamma\) in \(\mathbb R^3\), and given a function \(H : \mathbb R^3 \to \mathbb R\), find a surface \(M\) such that \(\partial M = \gamma\) and the mean curvature of \(M\) at \(p\) equals \(H(p)\) for all \(p \in M\). On this problem, there are two well-known results due to E. Heinz and S. Hildebrandt, respectively. One is that if \(\gamma\) is a circle in \(\mathbb R^3\) of radius \(R\) and if \(H_0 \geq 1/R\), then there does not exist a surface of constant mean curvature \(H_0\) bounding \(\gamma\). The other is that if \(\gamma\) is a Jordan curve in \(\mathbb R^3\) and if \(H : \mathbb R^3 \to \mathbb R\) is a function satisfying \(\| H\|_\infty \| \gamma\|_\infty \leq 1\), then there exists a surface of prescribed mean curvature \(H\) bounding \(\gamma\). To handle the conformality condition, one can consider the more standard Dirichlet problem instead of an analytical equation similar to \((P_0)\): \[ \begin{cases} \Delta u = 2 H(u) u_x \wedge u_y &\text{in }D\\ u = g &\text{on } \partial D, \end{cases} \tag \(D_H\) \] where \(g\) is a fixed continuous monotone parametrization of \(\gamma\). Using variational technique, truncation on \(H\) and maximum principle, Hildebrandt proved that if \(g \in H^{1/2}(S^1, \mathbb R^3) \cap C^0\) with \(\| H\|_\infty \| g\|_\infty \leq 1\), then the problem \((D_H)\) admits a solution. In section 5, the authors discuss the regularity of the \(H\)-equation (1) including a priori estimates of solutions of the following problem: Find a regular map \(u : {\overline D} \to \mathbb R^3\) such that \[ \begin{cases}\Delta u = 2 H(u) u_x \wedge u_y &\text{in } D\\ u_x|^2 - |u_y|^2 = 0 = u_x \cdot u_y &\text{in } D\\ u_{|_{\partial D}} \text{ monotone parametrization of } \gamma. \end{cases}. \tag \(P_H\) \] It is known due to Wente that if \(H\) is constant, then any weak solution \(u \in H^1(\mathcal U, \mathbb R^3)\) satisfying (1) is smooth. A generalized regularity result due to Bethuel is that if \(H \in C^\infty(\mathbb R^3)\) satisfies \(\| H\|_\infty + \| \nabla H\|_\infty \leq\infty\), then any solution \(u \in H^1(D, \mathbb R^3)\) to (1) with \(D = \mathcal U\) is smooth. The proof of this result involves the notion of Lorentz spaces which are borderline for Sobolev embeddings. Concerning a priori estimates of solutions to the \(H\)-equation, they stated that if \(u\) is a smooth solution satisfying the standard Dirichlet problem \((D_H)\), then \[ \| u\|_\infty \leq C \left(\| g\|_\infty + \| H\|_\infty \int_D |\nabla u|^2 + \left(\int_D |\nabla u|^2\right)^{1/2}\right). \] As an application of the a priori estimates above, one can obtain that if \(M\) is a compact surface in \(\mathbb R^3\), diffeomorphic to \(S^2\) and of mena curvature \(H\), then \[ \max_{p\in M} |H(p)| \geq C \frac{\text{diam}(M)}{\text{area}(M)}. \] In the last part of section 5, the authors gave some results on isoperimetric inequalities. Let \(u \in H^1_0(D, \mathbb R^3)\). Then \[ \left| \int_D u \cdot u_x \wedge u_y \right| \leq \frac{1}{\sqrt{32 \pi}} \left(\int_D |\nabla u|^2 \right)^{3/2}. \] This kind of isoperimetric inequality corresponds to the case of constant mean curvature \(H_0\). As far as the case of variable \(H\) is concerned, we can consider a generalized volume functional \[ V_H(u) = \int_D Q_H(u)\cdot u_x \wedge u_y, \] where \(Q_H : \mathbb R^3 \to \mathbb R^3\) is such that div\(Q_H = H\). Then if \(H \in L^\infty(\mathbb R^3)\), there exists a constant \(C_H\), depending only on \(\| H\|_\infty\), such that \[ |V_H(u)| \leq C_H \left(\int_D |\nabla u|^2 \right)^{3/2}. \] for every \(u \in H^1_0 \cap C^0\).
In section 6, the authors discuss the Rellich conjecture which states that in case of constant mean curvature \(H_0 \neq 0\), if \(\gamma\) is a Jordan curve such that \(\| \gamma\|_\infty |H_0| < 1\), then there exists a pair of parametric surfaces spanning \(\gamma\). Technically, the main dufficulty in showing the Rellich conjecture is to prove that the Dirichlet equation \((D_H)\) admits two solutions. The following result in this direction is due to Brézis and Coron. Let \(g \in H^{1/2} \cap C^0(\partial D, \mathbb R^3)\) and let \(H_0 \neq 0\) such that \(\| g\|_\infty |H_0| < 1\). If \(g\) is nonconstant, then the Dirichlet problem \((D_H)\) admits at least two solutions. To prove this result, we need several advanced notions on functional anlysis and variational techniques including the mountain-pass theorem and Palais-Smale sequence. In this section, they also characterize the solutions on \(\mathbb R^2\) of the problem \[ \begin{cases}\Delta u = 2 H_0 u_x \wedge u_y & \text{on } \mathbb R^2\\ \int_{\mathbb R^2} |\nabla u|^2 \leq \infty \end{cases}. \] Here \(H_0 \neq 0\) is a constant. In case \(H\) is nonconstant and \(D\) is a disk, the following result due to Bethuel and Rey can be proved. Let \(g \in H^{1/2} \cap C^0(\partial D, \mathbb R^2)\) be nonconstant and \(H_0 \neq 0\) be such that \(\| g\|_\infty |H_0| \leq 1\). Then there exists \(\varepsilon > 0\) such that for any \(H \in C^1(\mathbb R^3)\) satisfying \[ \| H - H_0\|_\infty < \varepsilon, \] the problem \((D_H)\) admits at least two solutions.
In the last section, the authors discuss the so called \(H\)-bubble problem. The analytical formulation of the \(H\)-bubble problem is the following: Find a nonconstant conformal function \(u : \mathbb R^2 \to \mathbb R^3\), smooth as a map on \(S^2\) satisfying \[ \begin{cases}\Delta u = 2 H(u) u_x \wedge u_y &\text{on } \mathbb R^2\\ \int_{\mathbb R^2} |\nabla u|^2 < \infty. \end{cases}\tag \(B_H\) \] In view of the variational method, one can detect solutions to \((B_H)\) as critical points of the energy functional \[ E_H(u) = \frac{1}{2} \int_{\mathbb R^2} |\nabla u|^2 + 2 \int_{\mathbb R^2} Q_H(u)\cdot u_x \wedge u_y, \] where \(Q_H : \mathbb R^3 \to \mathbb R^3\) is any vector field such that div\(Q_H = H\). Denoting by \({\mathcal B}_H\) the set of all \(H\)-bubbles and assuming \({\mathcal B}_H\) is nonempty, the authors considered wether the infimum is bounded by below and the infimum is taken by some \(H\)-bubbles. Finally, the authors mentioned a perturbed \(H\)-bubble problem.

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
58E12 Variational problems concerning minimal surfaces (problems in two independent variables)
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