Consider a bounded, connected and open subset $\Omega\subset \mathbb{R}^d$ and the Dirichlet Laplacian $-\Delta$ acting in $L^2(\Omega)$.

Then we know that the eigenvalues of $-\Delta$ form an increasing sequence of eigenvalues $0<\lambda_1<\lambda_2\leq \lambda_3\leq....$ and we have a corresponding orthonormal basis of eigenfunctions in $L^2(\Omega)$. Furthermore the eigenfunctions are smooth functions on $\Omega$.

If now $\Omega$ is sufficiently regular (e.g. smooth), then the eigenfunctions will vanish pointwise for all $x\in\partial \Omega$. I'm wondering if this is still true when no regularits assumptions are made? Do one has pointwise $\varphi(x)=0$, $\forall x\in\partial \Omega$ and all eigenfunctions $\varphi$?

Best regards,