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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.

##### MSC:
 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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