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,



Boundary regularity of Dirichlet Eigenfunction on bounded domains
In case this helps, the answer is yes for a Lipschitz domain.


From McLean's book *Strongly Elliptic Systems and Boundary Integral Equations*:


If $\Omega$ is a Lipschitz domain, there is a bounded trace operator $\gamma\colon H^s(\Omega) \to H^{s-1/2}(\partial\Omega)$ for any $1/2 < s < 3/2$. Moreover, if $1/2 < s \leq 1$, one has $$H^s\_0(\Omega) = \{u \in H^s(\Omega) : \gamma u = 0\}.$$


In particular, this implies that $H^1\_0(\Omega)$ is precisely the set of $H^1(\Omega)$ functions that vanish on the boundary.


