# Sobolev regularity for systems of elliptic boundary value problems

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!

• Have you looked at Morrey's classical book Multiple integrals in the calculus of variations ? Chapter 6 of that book seems to have what you need. – Liviu Nicolaescu Jul 28 '16 at 4:49
• Aaaaah I was afraid that Morrey might be the place to look! I always have such a fight with his notation when I try to read him, but encouraged by your suggestion that he indeed may have what I need, I'll gird my loins and fight through Ch 6! Thanks!! – Idempotent Jul 28 '16 at 16:37

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.

• Thank you! This looks like an excellent reference for the scalar case, but (unless I am overlooking something!) the vector case (by which I mean vector-valued functions that satisfy an elliptic BVP) is not treated here? – Idempotent Jul 27 '16 at 22:57
• Elliptic systems are discussed. – Michael Renardy Jul 27 '16 at 23:20
• I'm returning to say that in case anyone is reading this question, the book of Renardy and Rogers suggested above is excellent and very readable. It's also a huge help in reading Morrey, which though a classic and very complete, is a tough nut to crack for the modern reader (or at least this one!) who is used to more self-contained statements of results! – Idempotent Aug 1 '16 at 16:33
• I just want to assure you that less modern readers find Morrey's book hard to penetrate, too. – Michael Renardy Aug 1 '16 at 16:48
• Phew!!! I'm glad I'm not alone!! – Idempotent Aug 1 '16 at 18:48

I think the reason that you can hardly find reference about the vector valued functions is that you can always reduce your situation to scalar ones, since you can always use test function such that it is not zero only at the i-th position.

• That's true. But the i-th equation could still involve other components of the solution vector, because there could be a matrix of coefficients in the system that did some mix 'n' match. We'd have to think of these as coefficients in the i-th equation so as to regard it as an equation for a single scalar function representing just one component of the solution vector. My thinking is that this will let us prove some regularity results but probably not the strongest that may be possible, because we have likely made the coefficients less nice than the actual coefficients of the original system. – Idempotent Aug 6 '16 at 15:22