Regularity and Singularities in Problems on Domains With Boundary

Abstract

If Ω ⊂ Rⁿ is an open domain with smooth boundary, one considers the regularity of solutions to strictly hyperbolic equations of the form

$$ \rho (t,X,D)u - f(t,X)\,on\,R\times \Omega $$

in terms of the regularity of f, the restriction of u to a spacelike initial surface or to a region in the past which determines the values of u in the future, and the restriction of u to the boundary R ×∂Ω. A fundamental example is the following.