Let $P$ be a second order elliptic operator defined in a smooth bounded domain $\Omega \subseteq \mathbb{R}^n, n \geq 2$. $\lambda_0$ is the principal eigenvalue of the Dirichlet eigenvalue problem $$ \begin{aligned} & P u=\lambda u \quad \text { in } \Omega . \\ & u=0 \quad \text { on } \partial \Omega . \end{aligned} $$ In [1,2] Donsker and Varadhan generalized the variational formula for general second order elliptic operators with $C^{\infty}$ coefficients. Donsker and Varadhan proved that $ \lambda_0$ is given by $$ \lambda_0=\inf _{\mu \in \mathcal{M}} \sup _{u \in \mathcal{D}} \int_{\Omega} \frac{P u}{u} \mu(d x) . $$ where $\mathcal{M} \equiv \mathcal{M}(\Omega)$ is the space of all probability measures on $\bar{\Omega}$, and $\mathcal{D}$ denotes the set of all positive functions $u \in C^{\infty}\left(\mathbb{R}^n\right)$ for each of which there exist constants $c_1$ and $c_2$ such that $0<c_1 \leq u(x) \leq c_2<\infty$ for all $x \in \mathbb{R}^n$. The proof is based on strongly continuous semigroups theory and was motivated by a probability theory.
Besides, in [3], is was stated that the first eigenvalue is always real, but by the above formula, the space of test function is always strictly positive.
So for non self-adjoint linear elliptic operator, we have a real principal eigenvalue and a corresponding strictly positive eigenfunction? This seems to be a very strong conclusion. If this is true, is this still true on closed manifold?
[1]Donsker, Monroe D., and SR Srinivasa Varadhan. "On a variational formula for the principal eigenvalue for operators with maximum principle." Proceedings of the National Academy of Sciences 72.3 (1975): 780-783.
[2]Donsker, Monroe D., and SR Srinivasa Varadhan. "On the principal eigenvalue of second‐order elliptic differential operators." Communications on Pure and Applied Mathematics 29.6 (1976): 595-621.
[3]Protter, Murray H., and Hans F. Weinberger. "On the spectrum of general second order operators." (1966): 251-255.
