Weakly Elliptic Systems with Obstacle Constraints Part I — A 2 × 2 Model Problem

This note is intended to be the first in a series of papers treating linear elliptic systems of partial differential operators subject to obstacle type constraints. There is a large literature concerning solutions to linear and nonlinear elliptic systems of partial differential equations, but there seems to be much less work devoted exclusively to the understanding of solutions to such systems when they are subject to constraints. These constrained systems often take the form of a system of variational inequalities. Such problems have been treated, for example in [F1] and [HW] at least for strongly elliptic, i.e. Legendre-Hadamard elliptic, systems of variational inequalities. In this note we want to begin a study of a broader class of such systems, what we shall refer to as weakly elliptic systems — elliptic in the standard sense that the characteristic form of the principal part has no real zeros. One very important feature of this larger class is that the solution vector(s) can not in general, have the same degree of regularity in each component direction, as is generally the case for strongly elliptic systems. And as is generally well understood, solutions to variational inequalities only inherit a very limited amount of regularity from the data. For weakly elliptic linear systems, this inherited regularity is intimately tied up with certain algebraic structure considerations, considerations that do not appear when the obstacle constraints are removed.