The Dirichlet problem for prescribed curvature equations of \(p\)-convex hypersurfaces (Q6564481)

The author looks for a graphic hypersurface of prescribed curvature over a bounded domain \(\Omega \subset \mathbb{R}^{n}\), which is supposed to be given as the graph of a smooth function \(u:\rightarrow \mathbb{R}\) \N\[\NM_{u}=\{(x,u(x)):x\in \Omega \}\subset \mathbb{R}^{n+1}.\N\] \NThe problem is written as: Find a graphic hypersurface \(M_{u}\) whose principal curvatures satisfies the equation \N\[\N\mathcal{M}_{u}^{p}=\Pi_{1\leq i_{1}<\ldots <i_{p}\leq n}(\kappa_{i_{1}}+\ldots +\kappa_{i_{p}})=f(x,u,\nu)\N\] \Nin \(\Omega\), and the prescribed homogeneous boundary condition \(u=0\) on \(\partial \Omega\), where \(\kappa \lbrack M_{u}]=\{\kappa_{1},\ldots,\kappa_{n}\}\) are the principal curvatures of \(M_{u}\) with respect to the downward unit normal of \(M_{u}\), \(p\) is an integer with \(1\leq p\leq n\), \(\nu\) is the upward unit normal vector to the graphic hypersurface at \(X=(x,u(x))\) and \(f(x,z,\nu)>0\) is a smooth function defined on \(\overline{\Omega} \times \mathbb{R}\times \mathbb{S}^{n}\). \N\NThe first main result proves that if \(\Omega\) is a strictly convex bounded domain in \(\mathbb{R}^{n}\) with smooth boundary \(\partial \Omega\), \(f=f(x,z,\nu)\in C^{\infty}(\overline{\Omega}\times \mathbb{R}\times \mathbb{S}^{n})\) is a positive function with \(f_{z}\geq 0\), \(p\geq n/2\), and there is an admissible subsolution \(\underline{u}\in C^{2}(\overline{\Omega})\) satisfying \(\mathcal{M}_{\underline{u}}^{p}=f(x,\underline{u},\underline{\nu})\) in \(\Omega\) and \(\underline{u}=0\) on \(\partial \Omega\), where \(\underline{\nu}\) denotes the upward unit normal vector to the graphic hypersurface \(M_{\underline{u}}\) at \(\underline{X}=(x,\underline{u}(x))\), then there exists a unique admissible solution \(u\in C^{\infty}(\overline{\Omega})\) to the above problem. \N\NThe second main result proves that if \(\Omega\) is a bounded domain in \(\mathbb{R}^{n}\), and \(u\in C^{4}(\Omega)\cap C^{0,1}(\overline{\Omega})\) is an admissible solution to the above problem, then there are positive constants \(C\) and \(\beta\) which depend on \(n\), \(p\), \(\left\vert u\right\vert_{C^{1}}\), \(\inf f\) and \(\left\vert f\right\vert_{C^{2}}\) such that the second fundamental form \(h\) of \(Mu\) satisfies \N\[\N\sup_{\Omega}(-u)^{\beta}\left\vert h\right\vert \leq C,\N\]\Nas long as \(p\geq n/2\). \N\NThe author recalls and proves properties of the principal curvatures considered as the eigenvalues of the second fundamental form \(h\) of \(M\) with respect to the induced metric \(g\) on \(M\). He proves properties and lower bounds on the functions \(F(\kappa)=\Pi_{1\leq i_{1}<\ldots <i_{p}\leq n}(\kappa_{i_{1}}+\ldots +\kappa_{i_{p}})\) and \(\widetilde{F}=F^{1/C_{n}^{p}}\), and their first and second derivatives with respect to \(a_{ij}=\frac{1}{w}\gamma ^{ik}u_{kl}\gamma ^{lj}\), with \(\gamma ^{ik}=\delta ^{ik}-\frac{u_{i}u_{k}}{w(1+w)}\). He proves the interior estimate \N\[\N\sup_{\Omega}\left\vert D^{2}u\right\vert \leq C(1+\sup_{\partial \Omega}\left\vert D^{2}u\right\vert),\N\]\Nas long as \(p\geq n/2\), where \(C\) depends on \(n\), \(p\), \(\left\vert u\right\vert_{C^{1}}\), \(\inf f\) and \(\left\vert f\right\vert_{C^{2}}\), the boundary estimate \(\max_{\partial \Omega}\left\vert D^{2}u\right\vert\), if \(\Omega\) is a strictly convex bounded domain in \(\mathbb{R}^{n}\) with smooth boundary and \(u\in C^{3}(\overline{\Omega})\), and \(\sup_{\Omega}\left\vert Du\right\vert \leq C(1+\sup_{\partial \Omega}\left\vert Du\right\vert)\), if \(u\in C^{3}(\Omega)\cap C^{1}(\overline{\Omega})\) and \(f(x,z,\nu)\in C^{\infty}(\overline{\Omega}\times \mathbb{R}\times \mathbb{S}^{n})>0\) and \(f_{z}\geq 0\).