Note on a maximum principle and a uniqueness theorem for an elliptic-hyperbolic equation

Abstract

A maximum principle is proved for the function $\psi $ = $\int $[-2u$_{x}$u$_{y}$ dx + (Ku$_{x}^{2}$ - u$_{y}^{2}$) dy], where u is a solution of the equation of mixed type K(y)u$_{xx}$+u$_{yy}$ = 0 with K(y) $\gtrless $ 0 for y $\gtrless $ 0. The proof rests in showing that $\psi $ satisfies an elliptic equation for y > 0 and that it is a non-decreasing function of y for y $\leq $ 0. This maximum principle leads to a uniqueness theorem for the appropriate analogue to the Dirichlet problem for mixed equations under some conditions on the shape of the boundary curve. Very weak restrictions are imposed on K(y).