I am reading a paper which deals with the second variation of the domain functionals and applications . The following problem is solved . The eigen value $\lambda(t)$ which is characterised by the Rayleigh quotient : $$R(u,\Omega_t)= \frac{\int_{\Omega_t} |\nabla u|^2 dy + \alpha \oint_{\partial \Omega_t} u^2 dS}{\int_{\Omega_t} u^2 dy}$$ $\lambda(\Omega_t) = min\{R(v, \Omega_t) : v \in H^{1,2}(\Omega_t)\}$ The minimiser $u$ of the Rayleigh quotient satisfies : $$\Delta u + \lambda (\Omega_t) u = 0 \quad \mbox{in} \quad \Omega_t$$ $$\partial_{\nu_t} u + \alpha u = 0 \quad \mbox{in} \quad \partial \Omega_t$$ And $$g(u) = \lambda(\Omega_t) u$$ , $$G'(s) = g(s)$$
Now If i change the denominator of the Rayleigh quotient to be $$\left(\int_{\Omega_t} u^q dx \right)^{2/q}$$ i have gotten $$\Delta u + \lambda \mu (u) u^{q-1} = 0 \quad \mathtt{on} \quad \Omega_t$$ $$\partial_{\nu} u + \alpha u = 0 \quad\mathtt{on}\quad \partial \Omega_t$$ Therefore $$g(x) = \lambda(t) \mu(u) u^{q-1}$$ Where , $$\mu(u) = \left ( \int_{\Omega_t} u^{q-1} \, dx \right )^{2/q -1}$$ Am i right in saying that $$G(x) = \int_{\Omega_t} g'(x) \, dx = \lambda(t) \left( \int_{\Omega_t} u^{q-1} \, dx \right)^{2/q} $$. I further want to find the first variation , second variation and third variation of the eigenvalue $\lambda(t)$. Can someone please check and let me know if i am right till here ? I would be grateful if someone had patience to help me around .