One answer is given by Wong [1]. It states that eigenfunctions are analytic (bounded domain and Dirichlet boundary condition assumed). Using integration by parts we obtain for an eigenfunction corresponding to some eigenvalue \(\lambda>0\)

\begin{equation*} \lVert Du\rVert_{L^2(U)}^2=\lambda \lVert u\rVert_{L^2(U)}^2. \end{equation*}

Then it is easy to show that

\begin{equation*} \lambda \sum_{\lvert \alpha\rvert=k-1} \int_{U} \lvert D^{\alpha}u\rvert^2 \ge \sum_{\lvert \alpha\rvert=k} \lVert D^\alpha u\rVert_{L^2(U)}^2. \end{equation*}

Using this result and an appropriate Sobolev embedding we get an estimate for the \(L^{\infty}\) norm of each derivative. This proves first the smoothness and the estimate the analyticity of \(u\).

References Link to heading

1. W. Wong, Answer to “Why are Dirichlet eigenfunctions real-analytic?”, May. 15, 2014. [Online]. Available: https://math.stackexchange.com/a/795795 [Accessed: Nov. 28, 2024].