I am looking for an example of a pure second order uniformly elliptic operator
$L=\sum\_{i,j=1}^da\_{ij}(x)D\_{ij}$ in a bounded domain $\Omega$ (with Dirichlet boundary conditions, for example) having a non-real eigenvalue in $L^2(\Omega)$.
Under the above assumptions, the operator $L$ has a discrete spectrum and the principal eigenvalue is real; however it is non-symmetric. Note that if $a\_{ij}=a$ (in particular if $d=1$), then every eigenvalue is real since the measure $d\mu=a^{-1}(x)\, dx$ symmetrizes $L$.
Does anybody know such an example?



Non real eigenvalues for elliptic equations
Here is a construction. It elaborates from perturbation analysis of eigenvalues. However it starts from the situation of a non-simple eigenvalue.


So, let me start with the standard self-adjoint $L\_0=-\Delta$. I assume that the Dirichlet problem admits an eigenvalue $\lambda\_0$ of multiplicity $2$ exactly. I denote $(u,v)$ an orthonormal basis of the eigenspace. A well-known example is $\lambda\_0=5$ for the domain $K=(0,\pi)\times(0,\pi)$ and
$$u(x)=\frac2\pi\,\sin x\_1\sin2x\_2,\quad v=\frac2\pi\,\sin2x\_1\sin x\_2.$$


Let $M=\sum a\_{ij}D\_{ij}$ be given with bounded (smooth) functions $a\_{ij}$, and form $L\_\epsilon=L+\epsilon M$. Perturbation analysis tells us that $L\_\epsilon$ admits a stable plane $\Pi\_\epsilon$, which depends smoothly upon $|\epsilon|<\!<1$, and $P\_0={\rm vec}(u,v)$. In addition, the restriction of $L\_\epsilon$ over $\Pi\_\epsilon$ is similar to a matrix $C\_\epsilon$ such that on the one hand $C\_0=\lambda\_0I\_2$, and on the other hand
$$\left.\frac{d}{d\epsilon}\right|\_{\epsilon=0}C\_\epsilon=\begin{pmatrix} \langle Mu,u\rangle & \langle Mu,v\rangle \\ \langle Mv,u\rangle & \langle Mv,v\rangle \end{pmatrix}=:X\_0.$$
Suppose now that $M$ has been chosen so that
$$(\dagger)\qquad
( \langle Mu,u\rangle - \langle Mv,v\rangle )^2+4 \langle Mu,v\rangle \langle Mv,u\rangle <0.
$$
Then its eigenvalues are complex conjugated. Therefore, for $\epsilon\ne0$ small enough, $C\_\epsilon$ will have non-real eigenvalues. Since these are eigenvalues of the Dirichlet problem for $L\_\epsilon$, this provides an example.


There remains to find coefficients $a\_{ij}$ satisfying ($\dagger$). Let me just choose in my example (on the square $K$) $M=aD\_{12}$. The functions $uD\_{12}u-vD\_{12}v$, $uD\_{12}v$ and $vD\_{12}u$ are linearly independent. Thus there exists a function $a$ such that
$$\int\_Ka(uD\_{12}u-vD\_{12}v)\,dx=0,\qquad\int\_KauD\_{12}v\,dx=1,\qquad\int\_KavD\_{12}u\,dx=-1.$$
This ends the construction.


**Edit**. It is known that one-dimensional elliptic second-order operators have a real spectrum. Actually the eigenvalues are simple, therefore the construction described above cannot be implemented.


**Redit**. In the construction above, the eigenvalues depend smoothly upon $\epsilon$. There is no branching phenomenon. This happens because the unperturbed operator $L\_0$ is self-adjoint, thus diagonalisable.


