Rigidity of a surface relative to deformations satisfying nonlinear differential equations of first order (Q909869)

Let z(x) be a regular mapping of a region \(\Omega \subset R^ n\) on \(S\subset R^{n+1}\) satisfying the equation \(F(\dot z(x),z(x),x)=0\). The surface S is said to be F-rigid or F-rigid inside, if \(y=z\) is the unique solution of the problem \(F(\dot y,y,x)=0\) (for \(x\in \Omega)\), \(y(x)=z(x)\) (for \(x\in \partial \Omega)\) in the set of y: \(\Omega\Rightarrow R^{n+1}\) or y: \(\Omega\to S\), resp. The paper gives sufficient conditions on F, under which the surface is or is not F-rigid and F-rigid inside. The proofs are based on the implicit function theorem and Lyapunov-Schmidt splitting scheme. The interesting example of F- rigidity of a minimal surface is given.