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.

$\begingroup$ What is the role of the extra variable $t$? $\endgroup$– Liviu NicolaescuDec 19, 2015 at 10:49

$\begingroup$ it describe the time dependence $\endgroup$– Rym TouibiDec 19, 2015 at 10:54

$\begingroup$ In this case you have a whole family of operators. $\endgroup$– Liviu NicolaescuDec 19, 2015 at 11:18
1 Answer
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 $1$.

$\begingroup$ Thank you for your answer. In this kind of operator "divergence ooerator" we cannot make depent the coefficients on a time variable? $\endgroup$ Dec 19, 2015 at 11:49