Initial boundary value problem for the wave equation in a domain with a corner (Q690137)

The mixed problem for the wave equation in the domains \(\{(t,x,y,w)\mid t>0\), \(x>0\), \(y>0\), \(w\in\mathbb{R}^ 2\}\) and \(\{(t,x,y,z,w)\mid t>0\), \(x>0\), \(y>0\), \(z>0\), \(w\in\mathbb{R}^ 2\}\) is considered. An oblique boundary condition is given on the boundary \(x=0\), and on the boundary \(y=0\) or \(z=0\) Dirichlet or Neumann boundary conditions are given. The authors develop the energy inequality for the above problems with non- homogeneous boundary conditions. This leads to an existence and uniqueness theorem. This energy estimate, which is the same as the one for the mixed problem in smooth domains, is obtained by reducing the problem to the one for symmetric hyperbolic systems of first order with positive boundary conditions.