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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: $$ \begin{cases} -\Delta u = \lambda u,\;\text{ in } \Omega, \\[4pt] \displaystyle \int_{\Omega_1} u = 0, \; \text{ where }\Omega_1\subset \Omega, \;\mathrm{meas}(\Omega_1) >0. \end{cases} \tag{1*} $$

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

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