Analyticity of solutions for semilinear elliptic systems of second order

Abstract

This paper deals with systems $\Delta{\mathbf z}(u,v)={\mathbf h}({\mathbf z}(u,v),{\mathbf z}_u(u,v),{\mathbf z}_v(u,v))$ , $(u,v)\in S\subset\mathbb R^2$ , where the right hand side ${\mathbf h}$ is a $N$ -valued, real analytic function. We prove that a solution ${\mathbf z}:S\to\mathbb R^N$ of such a system can be continued across a straight line segment $I\subset\partial S$ , if one prescribe certain nonlinear, mixed boundary conditions on $I$ , which are assumed to be real analytic too. This continuation will be constructed by solving certain hyperbolic initial boundary value problems, generalizing an idea of H. Lewy. We apply this result to surfaces of prescribed mean curvature and to minimal surfaces in Riemannian manifolds spanned into a regular Jordan curve $\Gamma$ : Supposing analyticity of all data, we show that both types of surfaces can be continued across $\Gamma$ .