My question is about Sobolev estimates near the boundary for elliptic systems (equivalently, elliptic boundary-value problems for vector-valued functions). 


Note, results for the scalar case are easier to find, but it seems more difficult to find ones for the case when the solution is a vector-valued function.


I am interested in results that go something like this:


Suppose we have a second-order linear elliptic system on some domain in $\mathbb{R}^n$ with a smooth boundary. Suppose also that we have a solution with some degree of Sobolev regularity (i.e. the solution belongs to $H^s$ for some $s$). If the nonhomogeneous part of the equation and the boundary data also have some given levels of Sobolev regularity, then we can conclude that the solution actually has a higher level of Sobolev regularity. Not just in the interior (i.e. not on sets that are relatively compact in a domain which we assume is open), but actually up to the boundary.


If anyone can point me toward a reference, that would be great! Thank you!



Sobolev regularity for systems of elliptic boundary value problems
Tooting my own horn: You can start with Renardy and Rogers, An Introduction to Partial Differential Equations. You will find references to the original papers there.


