Eigenfunction of an uniformly elliptic second order operator Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^d$ and let $L$ be a uniformly elliptic second order partial differential operator:
  $$Lu(x,t)=-\sum_{i,j=1}^{d}{a_{ij}(x,t)u_{x_{i}x_{j}}(x,t)}+\sum_{i=1}^{d}{b_{i}(x,t)u_{x_i}(x,t)}$$
where $x\in\Omega$ and $t>0.$
My question is:
Does exist a positive eigenfunction of the corresponding homogeneous Dirichlet problem and an associated positive eigenvalue?under which conditions?
Could you provied me by some references? Thanks.
 A: This happens for operators ```in divergence form'', i.e.,  Laplace operators associated to a Riemann metric on $\Omega$.  If the Riemann metric is described by the tensor $(g_{ij}(x))_{1\leq i,j \leq d}$ satisfying the positivity condition: $\exists c>0$ $\newcommand{\bR}{\mathbb{R}}$
$$ \sum_{ij}g(ij)(x)\xi_i\xi_j\geq c\vert \xi\vert^2,\;\;\forall \xi\in\bR^d. $$
If $(g^{ij}(x))$ denotes the   inverse matrix
$$\bigl(\; g^{ij}(x)\;\bigr):=\bigl(\; g_{ij}(x)\;\bigr)^{-1},\;\;\forall x\in\Omega, $$
and $|g|$ the determinant
$$ |g|=\det  \bigl(\; g_{ij}(x)\;\bigr), $$
then the associated Laplacian  is $\newcommand{\pa}{\partial}$
$$ Lu=-\frac{1}{\sqrt{|g|}}\sum_{i,j}\pa_{x_i}\left(\sqrt{|g|}g^{ij}\pa_{x_j} u\right). $$
The  first nonzero eigenvalue of this operator is  given by
$$\lambda_1=\min\int_\Omega u(x)Lu(x)\sqrt{|g|} dx,\;\mbox{subject to the constraint}\;\int_\Omega u(x)^2\sqrt{|g|} dx=1,\;\;u\bigl|_{\pa \Omega}=0.$$
The minimizers of this problem are eigenfunctions  of $L$ corresponding to  $\lambda_1$. It is not difficult to see that if $u$ is a minimizer of this problem, the so is $|u|$. To prove that $|u|>0$  in the interior of $\Omega$ use the maximum principle. In particular, this also show that $\lambda_1$ has multiplicity $1$.
