Note on a Maximum Principle and a Uniqueness Theorem for an Elliptic-Hyperbolic Equation

A maximum principle is proved for the function psi = int [-2u_xu_y dx + (Ku_x² - u_y²) 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 <= 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).