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.



Eigenfunction of an uniformly elliptic second order operator
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 $1$.


