On microhyperbolic mixed problems (Q1176086)

The authors study a Dirichlet problem of the type \[ Pu(x,t)=f(x,t),\quad x\in\mathbb{R}^ N,\quad t>0,\quad u(x,0+)=g(x),\quad x\in\mathbb{R}^ N, \] in the class of hyperfunctions. Here \(P\) is a differential operator of \(m\)- th order of polynomial form in \(D_ t\) with analytical real coefficients and \(f\) is a mild hyperfunction. Using canonical extensions the problem can be reduced to the study of some local or global cohomology groups of the complex of sheaves. Then the problem is extended to the boundary conditions (e.g. at \(x_ 1=0\)) with additional conditions of Shapiro- Lopatonskii type and unique solvability and microlocalization of this problems are treated.