On the nonlinear Tricomi problem (Q1315059)

Let \(T\) be the Tricomi operator in \(G \subseteq \mathbb{R}^ 2\): \(T[u] = yu_{xx} + u_{yy}\). \(G\) is bounded by \(\Gamma_ 0\) in \(\{y>0\}\) and by the characteristic \(\Gamma_ 1\), \(\Gamma_ 2\) in \(\{y<0\}\). Under certain conditions for \(\Gamma_ 0\) existence and regularity results for generalized solutions of the problem: \(T[u] = f(x,y,u)\) in \(G\), \(u = 0\) on \(\Gamma_ 0 \cup \Gamma_ 1\); are proved. The results use known a priori estimates for the related linear problem. Among other things \(f\) has to satisfy the Carathéodory condition. Not included in the existence and regularity result is the case \(f(x,y,u) = u | u |^ \rho\), \(\rho \geq - 1/2\), which is discussed separately and for which existence is obtained.