A mixed problem for the wave equation in a domain with an angle (scalar case) (Q914064)

Mixed boundary value problems with homogeneous initial conditions for the homogeneous hyperbolic equation \[ \sum_{j+| \alpha | =r}a_{j_{\alpha}}D^ j_ t D^{\alpha}_ x=0\quad (D_ t=i \partial /\partial t;\quad D^{\alpha}_ x=D_ 1^{\alpha_ 1}...D_ n^{\alpha_ n},\quad D_ k=i \partial /\partial x_ k,\quad | \alpha | =\alpha_ 1+...+\alpha_ n) \] are investigated in the full scale of Sobolev spaces depending on the parameters s,\(\tau\in R:\) s characterizes the order of smoothness of the solution with respect to all variables, \(\tau\) the additional smoothness with respect to tangential variables. The smaller s and \(\tau\) are the more generalized the solution is. Theorems are proved from which one can conclude to which Sobolev spaces the solution belongs to. They also help for studying Green's function of the considered problem.