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

Abstract

A maximum principle is proved for the function ψ = J [ — 2u x u y dx + (Ku 2 x — u 2 y ) dy], where u is a solution of the equation of mixed type K(y)u xx + u yy = 0 with K(y) ≷ 0 for y ≷ 0. The proof rests in showing that iff 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).