Eindeutigkeitsaussagen für das Tricomi-Problem im \({\mathbb{R}}^ 2\) für eine Klasse nichtlinearer Gleichungen gemischten Typs (Q788909)

For the nonlinear equation \(L[u]:=k(y)u_{xx}+u_{yy}=g(x,y,u,u_ x,u_ y)\quad(sign k(y)=sign y)\) in G, bounded by a piecewise smooth curve \(\Gamma_ 0\) for \(y>0\) which intersects the line \(y=0\) at the points \(A(-1,0)\) and B(0,0) and for \(y<0\) by the characteristics \(\Gamma_ 1\) and \(\Gamma_ 2\) through the points A and B which intersect at the point C the uniqueness of boundary value problems \(i)\quad u|_{\Gamma_ 0\cup \Gamma_ 1}=\phi\) and \(ii)\quad d_ nu|_{\Gamma_ 0}=k(y)u_ xd_ y-u_ ydx=\phi ds\), \(u|_{\Gamma_ 1}=\psi\) is proved by the energy-integral method. To prove uniqueness theorems a growth condition for the function \(g_ u(...)\) in the hyperbolic part of the region is needed.