Characteristic boundary problems for the Liouville equation (Q1006983)

The authors study the equation \(U_{xy}=k\exp U,\) where \(k\) is a constant, under the following boundary conditions respectively: {\parindent=6mm \begin{itemize}\item[(1)] \(U\in C(\overline{D})\cap C^{1,1}(D)\), \(U(x,0)=\mu (x)\), \(x\in P=[0,a]\), \(U(0,y)=\nu (y)\), \(y\in Q=[0,b]\), and \(\mu (0)=\nu (0)\) here \(D=]0,a[\times ]0,b[\); \item[(2)] \(U\in C(\overline{D})\cap C^{0,1}(D\cup P)\cap C^{1,0}(D\cup Q)\cap C^{1,1}(D)\), \(U_{y}(x,0)=\mu _{1}(x)\), \(x\in P\), and \(U_{x}(0,y)=\nu _{1}(y)\), \(y\in Q\); \item[(3)] \(U\in C(\overline{D})\cap C^{0,1}(D\cup P)\cap C^{1,0}(D\cup Q)\cap C^{1,1}(D)\), \(U_{y}(x,0)+h_{1}(x)\exp U(x,0)=w_{1}(x)\), \(h_{1}\in C(P)\), \(h_{1}(x)>0\) and \(U_{x}(0,y)+h_{2}(y)\exp U(0,y)=w_{2}(y)\), \(h_{2}\in C(Q)\), \(h_{2}(y)>0\). \end{itemize}} They obtain formulas for solutions to problems(1), (2) and (3).