Regularity of Laplacian eigenfunctions in convex polygon. Suppose I have $\Omega$, a convex polygon with sides $\Gamma_i$. Suppose also that $u\in L^2(\Omega)$ is an eigenfunction of the Laplacian with Robin boundary conditions (understood in the trace sense, $u_n$ denotes normal derivative)
$$
a_iu_n-b_iu=0
$$
where $a_i,b_i$ are constants with $a_i^2+b_i^2=1$ WLOG. Can we say that $u$ lies in $H^2(\Omega)$? Can we say the stronger condition that $u$ and $u_n$ are in fact analytic on each edge separately? I have a feeling that the convexity is important but other than that I can't find anything in the literature on this. Thanks in advance!
 A: A good reference for this is Grisvard, Elliptic Problems in Nonsmooth Domains SIAM book Chapter 3 is titled Second-Order Elliptic Boundary Value Problems in Convex Domains. He first proves the estimate you want in convex domains $\Omega$ of class $C^2$ and shows that the constant in the main estimate does not depend on $\Omega$ but on $\lambda$. This allows him to get the same estimate for domains that are convex but not smooth.
 I don't know about analiticity although I suspect it holds.
A: The short answer to your question is:  no, convexity is not sufficient to guarantee $H^2$ regularity of such eigenfunctions.  The generality of your Robin boundary conditions allows, for example, homogeneous Dirichlet conditions on one side of a vertex, and homogeneous Neumann conditions on the other.  When this happens, even at a convex corner, the local behavior of the solution may be singular.
Although not a Polygon, consider the following example.  Let 
$\Omega$ be the top half of the unit disk, and impose homogeneous Dirichlet conditions on the top of the disk and on the positive x-axis, and homogeneous Neumann conditions on the negative x-axis.  You can use polar coordinates and separation-of-variables to determine all of the eigenfunctions, which are products of sinusoids in theta and (first-kind) Bessel functions in $r$.  More specifically, the eigenfunction for the smallest eigenvalue is  $J_{1/2}(z r) \sin(\theta/2)$, where $z$ is the first positive root of $J_{1/2}$; incidently, the smallest eigenvalue is $z^2.$
The relevant bit for your question is that this eigenfunction behaves like $r^{1/2}$ near the origin, so it is certainly not in $H^2$; it is in $H^{1+s}$ for any $s\lt1/2$.   Some eigenfunctions will be more regular than this---in fact some will be analytic for this problem---but there will always be some eigenfunctions that are not in $H^2$, regardless of how deep you go into the spectrum.
If we changed the domain to be the sector of the unit disk that is in the first quadrant, and imposed analogous boundary conditions, then all eigenfunctions are in $H^2$.
If you care for some references on regularity in the presence of corners, here are two books, and a paper:
Grisvard, P. Singularities in boundary value problems, Masson, 1992, 22, xiv+199
Grisvard, P. Elliptic problems in nonsmooth domains, Pitman (Advanced Publishing Program), 1985, 24, xiv+410
Wigley, N. M. Asymptotic expansions at a corner of solutions of mixed boundary value problems, J. Math. Mech., 1964, 13, 549-576
The second Grisvard book has been reissued in the SIAM "Classics in Applied Mathematics" series, so it may be easier to obtain than the other references.
The Wigley paper addresses precisely the problem that you consider.
