Eigenvalue estimate of a Neumann-like Laplacian in a Lipschitz domain

I have read quite some references on the smallest (for Dirichlet) and second smallest eigenvalues for the following Neumann eigenvalue problem: $$\begin{cases} -\Delta u = \lambda u,\;\text{ in } \Omega, \\[4pt] \dfrac{\partial u}{\partial n} = 0. \end{cases}$$

Notice for above problem, its eigenpairs also serve as the eigenpair for the following eigenvalue problem: $$\begin{cases} -\Delta u = \lambda u,\;\text{ in } \Omega, \\[4pt] \displaystyle \int_{\Omega} u = 0. \end{cases} \tag{1}$$ I am curious if there is any dedicated literature on the estimate of the smallest eigenvalue of problem (1). Or even the following more general eigenvalue problem: $$\begin{cases} -\Delta u = \lambda u,\;\text{ in } \Omega, \\[4pt] \displaystyle \int_{\Omega_1} u = 0, \; \text{ where }\Omega_1\subset \Omega, \;\mathrm{meas}(\Omega_1) >0. \end{cases} \tag{1*}$$

• These operators are not self-adjoint, so perhaps the eigenvalues won't be real. – Christian Remling Nov 12 '16 at 18:48