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

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!