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Dirichlet problem for space-like hypersurfaces with prescribed scalar curvature in \(\mathbb R^{n,1}\). (English) Zbl 1043.53027
In this paper under review, the author proves a Dirichlet problem for space-like hypersurfaces with prescribed scalar curvature in Minkowski space. Let \({\mathbb R}^{n,1}\) be the Minkowski space \(\displaystyle{{\mathbb R}^{n,1} = \bigl({\mathbb R}^{n+1}, \sum_{i=1}^n dx_i^2 - dx_{n+1}^2 \bigr)}\) with the canonical coordinates \((x_1, \dots, x_{n+1})\). Let \(\Omega\) be a smooth bounded domain in \({\mathbb R}^{n}= \{x_{n+1} = 0\}\) and let \(u\) be a smooth function defined on \(\overline{\Omega}\) satisfying \(\displaystyle{\sup_{\overline{\Omega}} | Du| < 1}\). Such a function \(u\) is called a spacelike function. Let \(M\) be the graph of the function \(u\), i.e., \(M=\{(x, u(x)): x \in \overline{\Omega}\}\) which is a spacelike hypersurface of \({\mathbb R}^{n,1}\). Setting \(\hat e_i= e_i + u_i e_{n+1}\), where \(e_i\) denotes the canonical basis of \({\mathbb R}^{n+1}\) and \(u_i\) denotes the partial derivative, the matrix, \(A(Du)\), of the induced scalar product with respect to the basis \(\{\hat e_i\}\) is given by \(A(Du)_{ij} = \delta_{ij} - u_iu_j\) and the inverse \(\displaystyle{A^{-1}(Du)_{ij} = \delta_{ij} + \frac{u_iu_j}{\omega^2}}\), where \(\omega = \sqrt{1-| Du| ^2}\). Then the matrix of the curvature endomorphism of \(M\) in the basis \(\{\hat e_i\}\) is \(\displaystyle{\frac{1}{\omega} A^{-1}(Du) D^2u}\) and the \(m\)-th order curvature of \(M\) at each point \(p\in M\) is defined by \[ {\mathcal H}_m (p) = \frac{1}{_nC_m} \sigma_m(\lambda_1, \dots, \lambda_n), \] where \(\lambda_1, \dots, \lambda_n\) are the principal curvatures of \(M\) at the point \(p\) and \(\sigma_m\) the \(m\)-th elementary symmetric function. Thus the first order curvature is just the mean curvature and the second order curvature is the scalar curvature of \(M\). One can thus define the \(m\)-th order curvature operator acting on the spacelike function over \(\Omega\) by \[ {\mathcal H}_m[u] = \frac{1}{_nC_m \omega^m} \sum_{I, J} A^{IJ}(Du) (D^2u)_{IJ}, \] where the summation is over the \(m\)-tuples and \(A^{IJ}(Du)\) and \((D^2u)_{IJ}\) are the minors of indices \(I, J\) of \(A^{-1}(Du)\) and \(D^2u\), respectively. We call a smooth function \(u\) defined on \(\overline{\Omega}\) admissible if it satisfies \[ \sup_{\overline{\Omega}} | Du| < 1\quad \text{and} \quad {\mathcal H}_k[u] > 0\quad(1\leq k\leq m). \] The author proves if \(\Omega\) is a smooth bounded domain in \({\mathbb R}^3\) and strictly convex, and \(H\) a smooth positive function on \(\overline{\Omega}\), then given a spacelike function \(\varphi\) which is strictly convex, the following Dirichlet problem \[ \left\{ \begin{aligned} {\mathcal H}_2[u] &= H \quad \text{in}\quad \Omega\\ u&=\varphi\quad \text{on}\quad \partial \Omega \end{aligned} \right. \] has a unique solution. In case \({\mathbb R}^4\), the author also shows that if \(H\) is constant, then the Dirichlet problem is also uniquely solvable. The proof relies on a standard continuity method which reduces it to a priori \(C^1\)- and \(C^2\)-estimates. These estimates imply \({\mathcal H}_m\) is uniformly elliptic. The author shows two types of \(C^2\)-estimates, i.e., normal second derivatives estimates and mixed (tangential-normal) second derivatives estimates on the boundary. And the maximum principle is also needed to complete the proof of the main result. The dimension restriction, \(n=3\) or \(n=4\) and \(m=2\), follows from the maximum principle and normal second derivatives estimates, whereas the \(C^1\) global estimate and the tangential-normal second derivatives estimate on the boundary can be obtained without any dimension restriction.
The author also conjectures that such a Dirichlet problem might be always solvable in any dimension \(n\) and any \(m\) if \(\Omega\) is a smooth convex bounded domain which has at least \(m-1\) positive principal curvatures at every boundary point.

53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
35J65 Nonlinear boundary value problems for linear elliptic equations
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
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