# smoothness of solution for second order elliptic problem

Hello all,

could someone point me to a reference that ties the smoothness of the solution $u$ to the classical elliptic problem

$\nabla \cdot ( q \nabla u ) = f \;,\; x \in \Omega$

$u = g \;,\; x \in \Gamma = \partial \Omega$

to the smoothness of $f$, $q$ and $g$?

$\Omega$ is a convex polygonal domain in $\Re^d$ with $d \in \{2,3\}$. The boundary $\Gamma$ is piecewise linear (can have corners, e.g., if $\Omega$ is the unit square).

I am particularly interested in the (minimal) smoothness requirements for the forcing and boundary data $f$ and $g$, such that $u \in {\cal H}^2(\Omega)$ (not just locally).

I went through Evans' book on PDEs but he assumes homogeneous boundaries and proves only local smoothness $u \in {\cal H}_{\rm loc}^s(\Omega)$ based on assumptions on the forcing $f$. My $g$ is generally nonzero.

Also, would the smoothness theory for the BVP above extend to a Helmholtz problem with a pure Neumann BC?

Thanks for any good pointers!

Kind regards, -- Mihai

• Is $\Gamma$ the boundary of $\Omega$? If so, what kind of regularity are you assuming for $\Gamma$? – Yakov Shlapentokh-Rothman Mar 1 '11 at 0:06
• have you looked at Gilbarg and Trudinger? – Willie Wong Mar 1 '11 at 0:11
• @Yakov Shlapentokh-Rothman: I updated the problem definition. @Willie Wong: thanks for the pointer, I will check it out today. – Mihai Mar 1 '11 at 17:26
• For an example of things that can go wrong since $\Gamma$ is not smooth, see mathoverflow.net/questions/38054/… – Yakov Shlapentokh-Rothman Mar 2 '11 at 0:30
• Gilbarg and Trudinger assume smooth boundaries as far as I see. I am going through Grisvard right now to see what I can figure out from there. – Mihai Mar 2 '11 at 17:33

## 2 Answers

Grisvard's book is a standard reference for elliptic problems in domains with corners.

See http://en.wikipedia.org/wiki/Elliptic_regularity and references therein.