Interior regularity of solutions of the Tricomi problem (Q1900356)

Let \(\Omega\subset \mathbb{R}^2\) be a bounded domain with boundary \[ \partial\Omega= C_0\cup C_1\cup C_2\cup \{P_0, P_2, P_2\}, \] where \(C_0\) is an open smooth curve in \(y> 0\), which connects the points \(P_0= (x_0, 0)\) and \(P_1= (x_1, 0)\). \(C_1\), \(C_2\) are characteristics of the equation \[ K(y) u_{xx}+ u_{yy}+ a_1(x, y) u_x+ a_2(x, y) u_y+ a_0(x, y) u= f(x, y), \] where \(a_i(x, y)\in C^\infty(\mathbb{R}^2)\) \((i= 0, 1, 2)\), \(y(K(y))> 0\), \(K'(y)> 0\) at \(y= 0\), described by \[ x= \pm \int^y_{t= 0} \sqrt{- K(t)} dt+ C \] which intersect at \(P_2\). The interior regularity of distributional solutions of the Tricomi problem is shown. It is of interest that the proof for local regularity at an interior point of \(\Omega\) requires that \(C_1\) is a characteristic. A similar result in case of the Frankl problem is given.