Let $\varepsilon$ a small positive real and $\Omega_\varepsilon$ an open set of $\mathbb{R}^2$ such that $$\Omega_\varepsilon=\{(x,y) \mid x = R(\theta,\varepsilon) \cos(\theta)\quad y=R(\theta,\varepsilon) \sin(\theta)\}$$ where $R(\theta,\varepsilon)=1+\varepsilon f(\theta)$ such that $f$ is regular $2\pi$-periodic function.
Let $u_1^\varepsilon $ the first eigenfunction of Dirichlet problem
\begin{align} -\Delta u = \lambda u & & & \text{ in } \Omega_\varepsilon \\[8pt] \text{and } u=0 & & & \text{ in } \partial \Omega_\varepsilon \end{align}
1-Can I write $u_1^\varepsilon =A_0 J_0(\sqrt{\lambda }r)+\varepsilon \sum\limits_{\substack{n\in \mathbb{Z} \\}} A_n J_n(\sqrt{\lambda} r) e^{(in\theta)}$ ?
2-if so why ?
PS : $J_n$ is the bessel function of the first kind and $A_i \in \mathbb{C}$