Elliptic regularity for Robin boundary conditions

Suppose I have a (non-smooth) domain $\Omega$ on which I have a $H^1$ solution $u$ of a constant coefficient elliptic PDE $L$. Suppose also that $\Gamma$ is a smooth portion of the boundary $\partial\Omega$ and on a neighborhood of $\Gamma$ I prescribe Robin boundary conditions $$a\frac{\partial u}{\partial n}+bu=g$$ where $a,b$ are constants (not both zero but $a=0$ corresponds to Dirichlet BCs and $b=0$ to Neumann BCs) and $g$ is some function. My question is, given some sort of regularity of $g$, can we say anything about regularity of $u$ up to the boundary? So far, everything I can find deals only with the Dirichlet case. Anyone know any references for the general case?

• The PhD thesis of R. Nittka contains regularity estimates for Robin boundary conditions; you can find the thesis here: dx.doi.org/10.18725/OPARU-1790. The estimates in the thesis are quite general and also work for some non-linear problems. Maybe one of the results contains your problem as a special case? – Jochen Glueck Sep 26 '17 at 8:25
• That's close but doesn't cover what I have in mind. – Mathmo Oct 14 '17 at 19:56