What kind of regularity do we generally have for weak solutions to the Dirichlet problem?
$$(\Delta+\lambda)u=0 \textrm{ in }U$$
$$u=0 \textrm{ on }\partial U $$
where $U$ is a planar domain with Lipschitz boundary?
I have been told that such domains are nice enough to guarantee classical solutions, but I have not found any references for this.
If $\lambda$ is negative, the problem is $H^1(U)$-elliptic, and an $H^1(U)$ solution exists and is unique. If $U$ is convex or $C^2$, this solution is $H^2(U)$. If $\Omega$ has some reentrant corner, you can expand the solution in singular functions which vanish away from the reentrant corner. A good reference may be
If $\lambda$ is positive, you may run into trouble if $\lambda$ is an eigenvalue of the Laplacian. Otherwise, the same as above apply (more or less). For this case, you could have a look at