×

On some boundary value problems for an ultrahyperbolic equation. (English) Zbl 0914.35073

In the space of variables \(x_1,x_2,y_1\) and \(y_2\) we consider the ultrahyperbolic equation \[ u_{y_1y_1}+ u_{y_2y_2}- u_{x_1x_1}-u_{x_2x_2} =F.\tag{1} \] Denote by \(D:-y_1<x_1<y_1\), \(0<y_1<+\infty\), a dihedral angle bounded by the characteristic surfaces \(S_1:x_1-y_1=0\), \(0\leq y_1<+\infty\), and \(S_2:x_1+y_1=0\), \(0\leq y_1<+\infty\), of equation (1). We shall consider a characteristic problem formulated as follows: in the domain \(D\) find a solution \(u(x_1,x_2, y_1,y_2)\) of equation (1) by the boundary conditions \(u|_{S_i}=f_i\), \(i=1,2\), where \(f_i\), \(i=1,2\), are given real functions on \(S_i\) and \((f_1-f_2)|_{S_1\cap S_2}=0\).

MSC:

35L20 Initial-boundary value problems for second-order hyperbolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
PDF BibTeX XML Cite
Full Text: EuDML EMIS