# Inequalities concerning principal eigenvalues of Laplacian with different boundary conditions

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) and (2)).

$\textbf{Update:}$ Inequalities between Neumann and Dirichlet problems have been intensively studied for much more general setting e.g. Laplace-Beltrami operator(see, Payne and Friedlander) which may or may not be pertinent to this question.