**My situation**:
I am currently trying to understand [Uhlenbecks results](https://link.springer.com/article/10.1007%2FBF01947069?LI=true) 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](https://mathoverflow.net/questions/281375/principal-symbol-for-non-linear-differential-operators) 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](https://faculty.math.illinois.edu/~palbin/AnalysisOnMfds/LectureNotes.pdf) 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"](http://www.springer.com/de/book/9783540699156) 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.



Resources on Elliptic Boundary Value Problems on manifolds
As an addition to Hadrian's answer, for boundary value problems there are as you mention some issues with the boundary and choosing the correct boundary conditions.
As a rule of thumb, people want to have a theorem of the kind "ellipticity implies fredholmness". Thus, for bvps you have to add a notion of ellipticity at the boundary. This is also called the Shapiro-Lopatinskii condition. Roughly speaking you define an operator-valued symbol at the boundary and require that it is invertible.


Reference include: Hörmander 3 (Chapter XX.1) this is somehow the standard viewpoint, "A short introduction to Boutet de Monvel's calculus" by Elmar Schrohe (if you want to consider pseudodifferential boundary value problems), and the book by Egorov and Schulze (they have a readable introduction into elliptic bvp, but most of the book is really tough to read).


