Eigenvalue variations for the Neumann problem (Q5938876)

Let \((M,g)\) be a smooth Riemannian manifold of dimension \(m\). Let \(\Omega_\varepsilon\) be a smooth 1-parameter family of compact submanifolds of the ambient manifold \((M,g)\) which have smooth boundaries. Let \(v_0\) be the normal variation or speed of the perturbation when \(\varepsilon=0\). Impose Neumann boundary conditions to define the associated Laplacian and let \(\lambda(\varepsilon)\) be an eigenvalue. Let \(u_0\) be the associated eigenfunction when \(\varepsilon=0\). It is assumed \(\lambda(\varepsilon)\) varies smoothly; the first variation is then computed and shown to be NEWLINE\[NEWLINE\lambda'(0)=\int_{d\Omega_0} (|\nabla_{\partial\Omega_0}u_0|^2 -\lambda_0u_0^2)v_0 d\text{vol}.NEWLINE\]