Smoothness of solutions of second-order hyperbolic equations with discontinuous coefficients (Q1177812)

The author investigates the standard hyperbolic problem \[ u_{tt}=\sum^ n_{i,j=1}a_{ij}(x,t)u_{x_ ix_ j}+\sum^ n_{i=1}a_ i(x,t)u_{x_ i}+a(x,t) u+f, \] where the coefficients are smooth except across an \(n-1\)-dimensional surface. Further, the coefficients \(a_{ij}\) satisfy a uniform ellipticity condition. The Dirichlet problem is considered, but the author indicates that the results extend to the Neumann problem. Making minor assumptions on the function \(f\), and assuming zero initial conditions, the author derives several regularity results for the generalised (or weak) solution of the problem. Examples of the regularity shown are Hölder spaces and weighted Sobolev spaces.