These lecture notes provide aself-contained introduction to regularity theory for elliptic equations andsystems in divergence form. After a short review of some classical results oneverywhere regularity for scalar-valued weak solutions, the presentationfocuses on vector-valued weak solutions to a system of several coupledequations. In the vectorial case, weak solutions may have discontinuities andso are expected, in general, to be regular only outside of a set of measurezero. Several methods are presented concerning the proof of such partialregularity results, and optimal regularity is discussed. Finally, a shortoverview is given on the current state of the art concerning the size of thesingular set on which discontinuities may occur.The notes are intended for graduate andpostgraduate students with a solid background in functional analysis and somefamiliarity with partial differential equations; they will also be of interestto researchers working on related topics.
Preliminaries.-Introduction to the Setting.- The Scalar Case.- Foundations for the VectorialCase.- Partial Regularity Results for Quasilinear Systems.
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