×

The Dirichlet problem for a curvature equation of order m. (Russian) Zbl 0716.35027

The fully nonlinear equations studied in this paper are natural generalizations of both the mean curvature and Monge-Ampère equations. To be more specific consider \(\mu_ m=\sigma_ m/\left( \begin{matrix} n\\ m\end{matrix} \right)\) where \(\sigma_ m\) is the elementary symmetric polynomial of degree m in n variables. Any function u: \(\Omega\subset {\mathbb{R}}^ n\to {\mathbb{R}}\) defines a hypersurface in \({\mathbb{R}}^{n+1}\) with principal curvatures \(k_ i=k_ i(u)\), \(i=1,...,n\). Its m-th order curvature at the point (x,u) is \[ \mu_ m=\mu_ m(x,u)=\mu_ m(k_ 1(u),...,k_ n(u)). \] The m-th order curvature equation is related to the following geometric problem: find a hypersurface \(u=u(x)\) with given boundary \(u(x)=\phi (x)\) on \(\partial \Omega\) and prescribed m-th order curvature \(H_ m(x,u).\)
The novelty of this paper is twofold. It deals both with nonvanishing Dirichlet data and with non-necessarily convex domains \(\Omega\). The main results are theorems 4.1-4.4. They state that given some compatibility conditions between \(H_ m(x,u)\), \(\phi\) (x), and the geometry of \(\partial \Omega\) then there exists a unique solution \(u\in C^{k+2+\alpha}(\Omega)\) if \(m<n\), \(\partial \Omega \in C^{k+2+\alpha}\), \(\phi \in C^{k+2+\alpha}(\partial \Omega)\), where \(k\geq 2\) and \(\alpha\in (0,1).\)
The method of proof is based on a variant of Leray-Schauder method. This requires a priori \(C^ 2({\bar \Omega})\) estimates. These are obtained successively from \(C^ 1(\Omega)\) estimates by a clever use of a variant of maximum principle. Such a principle exists basically due to the compatibility conditions. In the appendix it is mentioned a recent result due to N. Trudinger which shows that these compatibility conditions are almost necessary.
Reviewer: L.Nicolaescu

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35B45 A priori estimates in context of PDEs
PDFBibTeX XMLCite