Suppose $\Omega\subset\mathbb R^2$ is a bounded simply connected domain with sufficiently smooth boundary. Consider the following three BVPs (respectively with Dirchlet, Neumann and certain non-local boundary condition):
$$(I):\, \begin{cases}  \Delta u= \lambda u,\,\,\,\text{in} \,\,\Omega  & \\ 
u|_{\partial\Omega}=0, & 
\end{cases} $$   

$$(II):\, \begin{cases}  \Delta v=\lambda' v ,\,\,\,\text{in} \,\,\Omega& \\ 
\frac{\partial v}{\partial n}|_{\partial\Omega}=0,
\end{cases} $$

$$(III):\,\begin{cases}  \Delta w=\lambda'' w,\,\,\text{in}\, \Omega \\ 
-\frac{1}{2}w(x)-\frac{1}{2\pi}\int_{\partial\Omega}\frac{\partial}{\partial n_y}\ln|x-y|w(y)dS_y+
\frac{1}{2\pi}\int_{\partial\Omega}\ln|x-y|\frac{\partial w(y)}{\partial n_y}dS_y=0, & \mbox{} x\in\partial\Omega.
\end{cases} $$
$\textbf{Question:}$ How are the principal eigenvalues of $(I)$ and $(II)$ compared to the principal eigenvalue of $(III)$? In case of $\Omega=\mathbb{D}$(=unit disk) eigenvalues can be  explicitly computed and the principal eigenvalue of $(III)$ is equal to the principal eigenvalue of $I$.
   
$\textbf{Context:}$ The non-local boundary condition in $(III)$ has to do with eigenvalues of the logarithmic potentials (see, [(1)][1] and [(2)][2]).  
 
$\textbf{Update.}$ Inequalities between Neumann and Dirichlet have been intensively studied for much more general setting i.e. Laplace-Beltrami operator(see, [Payne][3] and [Friedlander][4]) which may or may not be pertinent to this question.  


  [1]: http://arxiv.org/pdf/1503.08390v2.pdf
  [2]: http://qjmath.oxfordjournals.org/content/43/4/387.full.pdf+html
  [3]: http://onlinelibrary.wiley.com/doi/10.1002/sapm1960391155/abstract
  [4]: http://link.springer.com/article/10.1007/BF00375590