$L=\sum_{ij}a_{ij}(x)D_{ij}$ with real continuous coefficients $a_{ij}=a_{ji}$ (no lower order terms) and Dirichlet b.c on a bounded domain (say the ball) having some non-real eigenvalue?
Martin Berggren
A wild guess could be that you could get non-real eigenvalues by setting some weird boundary conditions, say oblique ones. The operator associated with a variational form of the problem will then be non symmetric, even if the leading term is something standard like the Laplacian.
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