General Boundary Value Problems for Differential Equations of Sobolev Type

Abstract

An $L_p $ theory $(1 < p < \infty )$ of existence and regularity of solutions of the partial differential equation $(1 - \gamma \mathcal{M}(t)){{\partial u} / {\partial t}}) - \mathcal{L}(t)u = f$ satisfying general boundary conditions is given. For each t, $\mathcal{M}(t)$ is a linear elliptic partial differential operator in the space variables, $\mathcal{L}(t)$ is a linear differential operator whose order does not exceed that of $\mathcal{M}(t)$ and $\gamma $ is a nonzero complex constant.