Local existence of solutions for the mixed problem of fully nonlinear partial differential equations (Q1891317)

The author proves the short-time existence of solutions of the mixed initial-boundary value problem for fully nonlinear hyperbolic equations. The Cauchy problem for such equations has been solved by \textit{P. Dionne} [J. Anal. Math. 10, 1-90 (1962; Zbl 0112.323)] by differentiating in both the space and time directions to obtain an equivalent quasilinear hyperbolic system to which standard iteration techniques can be applied. The same procedure fails when applied to the mixed problem because it is not possible to give proper boundary conditions for the system so obtained. Here the author shows, essentially, that by differentiating twice in the time direction only, an equivalent system with proper boundary conditions is obtained. This problem can then be solved by using energy estimates for linear hyperbolic equations satisfying a mixed boundary condition together with a suitable iteration scheme.