Existence for a characteristic boundary value problem for an equation of mixed type in three‐dimensional space

The equation of mixed typeWith k ( x 3 ) = sign x 3 | x 3 | m , m > 0, d ϵ C 1 (Ḡ), x = ( x 1 , x 2 , x 3 ), is considered in the threedimensional region G which is bounded by the surfaces: a piecewise smooth surface Γ 0 lying in the half‐space x 3 > 0 which intersects the plane x 3 = 0 in the unit circle, and for x 3 < 0 by the characteristic surfacesWe prove existence of a generalized solution for the characteristic boundary value problem: Lu = f in G , u Γ0∪Γ1 = 0. The result is obtained by using a variant of the energy‐integral method.