Regularity of Laplacian eigenfunctions in convex polygon.

Question

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!

Answer 1

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.