I have the following problem, \begin{equation} u_{tt} - \Delta u = 0, \, \, \text{with } x \in R \subset \mathbb{R}^N, \end{equation} \begin{equation*} u = 0 \, \, \text{at } \partial R. \end{equation*} The natural frequencies of this system, $\omega(\epsilon)$, depend on the region $R$ and we can obtain them by separation of variables: $u(x,t; \epsilon) = U(x;\epsilon) e ^{i \omega(\epsilon)t}$. The region is $R = R(\epsilon)$, and $\omega^0$ a natural frequency for $\epsilon = 0$, with unique normal mode $u^0(x)$, and $0<\epsilon \ll 1$. The boundary $\partial R (\epsilon)$ can be described in terms of $\partial R (0)$: for all $x \in \partial R (0)$ we define $x^1 = x + \epsilon b(x) \mathbf{n}(x)$, where $\mathbf{n}(x)$ is the exterior normal vector at $\partial R(0)$ in $x$ and $b(x)$ is a function which is known. I want to determine $\omega^1$ which is the second order in the asymptotic expansion of the frequency, $\omega(\epsilon) \sim \omega^0 + \omega^1 \epsilon$ ($\epsilon \to 0+$).
What I have done is to look for $2\pi$-periodic solutions, $u(x,t+2\pi) = u(x,t)$: $u(x,t;\epsilon) = U(x;\epsilon) e^{i \omega(\epsilon) t}$. So the spatial problem satisfies the Helmholtz equation, \begin{equation*} \Delta U + \omega^2 U = 0, \quad x \in R(\epsilon), \end{equation*} \begin{equation*} U = 0, \quad \text{en } x \in \partial R(\epsilon). \end{equation*} I have expand the boundary condition, \begin{equation*} 0 = u(x^1;\epsilon) = u(x+ \epsilon b(x) \mathbf{n};\epsilon) = u(x;\epsilon) + \epsilon b(x) \mathbf{n} \nabla u(x;\epsilon) + O(\epsilon^2), \end{equation*} \begin{equation*} 0 =U(x;\epsilon) + \epsilon b(x) \mathbf{n} \nabla U(x;\epsilon) + O(\epsilon^2). \end{equation*} And also I have expand in powers of $\epsilon$ $U(x;\epsilon)$ and $\omega(\epsilon)$, $U(x;\epsilon) = U^{(0)}(x) + \epsilon U^{(1)}(x) + O(\epsilon^2)$, $\omega(\epsilon) = \omega^{(0)} + \epsilon \omega^{(1)} + O(\epsilon^2)$, so finally I reach the following hierarchy of equations, \begin{equation*} \epsilon^0 : \begin{cases} U^{(0)} \omega^{(0)^2} + \Delta U^{(0)} = 0, \quad x \in R(0)\\ U^{(0)} = 0, \quad \text{ en } x \in \partial R(0). \end{cases} \end{equation*} \begin{equation*} \epsilon : \begin{cases} U^{(1)} \omega^{(0)^2} + \Delta U^{(1)} = -2 \omega^{(0)} \omega^{(1)} U^{(0)}, \quad x \in R(\epsilon)\\ U^{(1)} = - b(x)\mathbf{n} \nabla U^{(0)} , \quad \text{ en } x \in \partial R(\epsilon). \end{cases} \end{equation*} So once I have found the $\epsilon^0$ order, if I want to determine the correction of the frequency $\omega^{(1)}$ I apply the solvability condition. The operator $\mathcal{L} = \Delta + \omega^{(0)^2}$ is selfadjoint under a certain scalar product. Therefore, if the $\epsilon$ order equation (non-homogeneous equation) have a non-trivial solution, the image of $\mathcal{L}$, $-2 \omega^{(0)} \omega^{(1)} U^{(0)}$, needs to be orthogonal to its kernel, generated by $U^{(0)}$ under a certain scalar product, \begin{equation*} \int_R -2 \omega^{(0)} \omega^{(1)} U^{(0)^2} dR - \int_{\partial R} b(x) U^{(0)} \frac{\partial U^{(0)}}{\partial \mathbf{n}} = 0. \end{equation*} I do not know if the solvability condition for the boundary is correct or if I can do something else.