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

Abstract

AbstractThe equation of mixed typemagnified imageWith k(x3) = sign x3|x3|m, m > 0, dϵC1(Ḡ), x = (x1, x2, x3), is considered in the threedimensional region G which is bounded by the surfaces: a piecewise smooth surface Γ0 lying in the half‐space x3 > 0 which intersects the plane x3 = 0 in the unit circle, and for x3 < 0 by the characteristic surfacesmagnified imageWe prove existence of a generalized solution for the characteristic boundary value problem: Lu = fin G, uΓ0∪Γ1 = 0. The result is obtained by using a variant of the energy‐integral method.