The exixtence of generalized solutions for a class of linear and nonlinear equations of mixed type

Abstract

In this paper we deal with the question of existence and uniqueness of the generalized solutions for a class of linear and nonlinear equations of the mixed type. In particular we consider $L(u)≡ k(y) u_{xx} + u_{yy} = f (x,y,u)$ in a simply connected region G , where $k(y)≥ 0$ for $y≥ 0$ and $k(y)< 0$ for $y < 0$. G is bounded by the curves Γ₀,Γ₁,Γ₂.Γ₀$ is a piecewise smooth curve lying in the half plane $y > 0$ which intersects the line $y= 0$ at the points $P(- 1,0)$ , $Q(0,0)$; $Γ₁$ is a piecewise smooth curve through P in $y < 0$ which meets the characteristics of the above operators issued from Q al the point R and $Γ₂$ consists of the portion RQ of the characteristic through Q. We assume that $Γ₁$ either lies in the characteristic triangle formed by the characteristics through P and Q (Frankl Problem) or coincides with the characteristics through P (Tricomi Problem). We seek sufficient conditions for the existence and uniqueness of generalized solutions of the boundary problem $$L(u) = f(x,y,u)\;\textrm{in} G,u_(Γ₀\cupΓ₁}=0.$$