On the principal eigenvalue of indefinite elliptic problems (Q1077624)

We consider the linear elliptic system \[ \ell \vec u=-\Delta \vec u+2\sum b_ j D_ j \vec u+A \vec u=\lambda W \vec u \] in a bounded subdomain \(\Omega \subset R^{\eta}\). We assume that both \(\ell\) and W are indefinite. Under suitable conditions on \(\Omega\) and the coefficients we give conditions sufficient for \(\ell \vec u=\lambda W \vec u\) to have positive eigenvectors - or not - and both upper and lower estimates on the corresponding eigenvalues. Our main tools are the theory of positive operators, maximum principle arguments, and an identity which relates the spectrum of \(\ell\) to the spectrum of symmetric expressions.