Second-order linear elliptic systems on the complex plane

Abstract

Let ${\cal S}$ denote the general second order linear differential operator on the complex plane. It is a well-known fact that the Dirichlet boundary value problem for ${\cal S}$ on a bounded domain $\Omega$ with a smooth boundary, is not always well-posed even when ${\cal S}$ is elliptic.

This phenomenon led to a homotopic classification of elliptic systems. B. Bojarski, a pioneer in this subject, showed that the family of elliptic operators forms an open set in $\doubc\sp6$ with exactly six components.

It follows from his classification that the only components where the Dirichlet problem may be well-posed are the ones represented by the Laplacian or its complex conjugate.

We denote by ${\cal E}(\Delta)$ the component consisting of elliptic operators that are deformable to the Laplacian. The objective of this dissertation is to establish the Fredholm alternative for the equation: ${\cal SW}(z)=g(z),{\cal S}\in{\cal E}(\Delta).$ Using the complex Hilbert transforms, the problem reduces to showing that Lw(z) = g(z) satisfies the Fredholm alternative, where L is an integral operator and $w\in L\sp{p}(\doubc)$.

Assuming first that the coefficients of ${\cal S}$ are constant functions, we show that L is in fact invertible. In case the coefficients are only measurable, we construct an example of an operator with a non-trivial kernel.

If the coefficients are uniformly continuous functions on $\doubc,$ we show that L may be a Fredholm operator with index equal to zero for p close to 2.

The main result of this paper generalizes the latter as follows: we assume that the coefficients are functions of vanishing mean oscillation and we show that L is Fredholm with index zero for all $p\in(1,\infty).$