Eigenfunction of Dirichlet Laplacian on smooth domain in $\mathbb{R}^n$

Question

I was reading about eigenfunctions of the Dirichlet Laplacian on bounded domains $\Omega \subset \mathbb{R}^n$. It seems that such eigenfunctions are real analytic in the interior of $\Omega$ and smooth up to the boundary $\partial \Omega$ if the boundary is smooth. If I understand this correctly, this means that any eigenfunction $u_j$ (satisfying $-\Delta u_j = \mu_j u_j$) has a smooth extension slightly beyond $\partial \Omega$. I was wondering, could we define the extension in a way that even on and across the boundary, the pde $-\Delta u_j = \mu_j u_j$ is still satisfied? Thanks!