A self-adjoint problem for the wave equation in higher dimensions (Q810294)

Symmetric Green's functions are given for \(Lu\equiv u_{tt}-\Delta u=f\) in space dimensions of \(n=1,2\) and 3. In the case \(n=1\) the Green's function \[ G(x,t;\xi,\tau)=\frac{1}{4}H((t-\tau)\quad -\quad | x-\xi |)\quad +\quad \frac{1}{4}H((\tau -t)-| x-\xi |) \] corresponds to a selfadjoint boundary problem in the characteristic triangle \(\{\) (x,t): t-1\(\leq x\leq 1-t,0\leq t\leq 1\}\). This leads to the question of whether there exist regions and boundary conditions which generate selfadjoint boundary value problems for the wave equation in space dimensions \(n>1\).