Failure of Fredholm Solvability for Weakly Elliptic Systems

Abstract

It is known that the $L^p$ Dirichlet Problem for constant coefficient second-order systems satisfying the Legendre-Hadamard (strong) ellipticity condition is well posed in the upper half-space. We have already seen in Chapter that this may not be the case in the class of weakly elliptic scalar operators in the complex plane. As we shall see in this chapter, counterexamples exist for weak elliptic systems in all space dimensions. In fact, the failure of the corresponding $L^p$ Dirichlet Problems to be well posed is at a fundamental level, in the sense that as we shall see momentarily, there exist weakly elliptic systems in ${\mathbb {R}}^n$ with $n\ge 2$ for which the $L^p$ Dirichlet Problem is not even Fredholm solvable. The manner in which this ties up with our earlier work in Chapter 1 is that we shall look for such a pathological weakly elliptic system in the class of those which fail to possess a distinguished coefficient tensor.