Existence and uniqueness theorems are proved for boundary value problems with trihedral corners and distinct boundary conditions on the faces. Part I treats strictly dissipative boundary conditions for symmetric hyperbolic systems with elliptic or hidden elliptic generators. Part II treats the Bérenger split Maxwell equations in three dimensions with possibly discontinuous absorptions. The discontinuity set of the absorptions or their derivatives has trihedral corners. Surprisingly, there is almost no loss of derivatives for the B\'erenger split problem. Both problems have their origins in numerical methods with artificial boundaries