**My situation**:
I am currently trying to understand Uhlenbecks results on the Yang Mills equation. One of the most common notions in this paper is that of an **elliptic system** or an **elliptic boundary value problem**. However, as far as my knowledge goes I only know *ellipticity* as a condition on the leading coefficients of a second order linear differential operator. Or, a bit more general, on pseudodifferential operators acting on $\mathbb{R}^n$. I already asked another question on principal symbols for non-linear operators, which (partly) explains what ellipticity should mean for these. Nevertheless, I do not see, what role the boundary plays in the definition of an elliptic BVP.

**Question**: Are there any resources which define the terminology of an *elliptic boundary value problem* for non-linear operators of arbitrary order? Best, even on general domains, such as manifolds with boundary?

**Remarks:** There are some references which touch the kind of exposition I am looking for: I found this set of lecture notes on *Linear Analysis on Manifolds* which comes close to what I am looking for, but leaves out non-linear problems as well as boundary conditions. Furthermore "Multiple Integrals in the Calculus of Variations" by Morrey Jr. seems to be an often cited reference as well. And last but not least, the work of Agmon, Douglas an Nirenberg seems to be quite foundational.