Elliptic regularity theory. A first course (Q896540)

These lecture notes provide a self-contained introduction to regularity theory for elliptic equations and systems in divergence form. The first part of this monograph is devoted to some classical results about everywhere regularity for scalar-valued weak solutions. The seminal results due to DeGiorgi and Moser are presented. In the vectorial case, weak solutions may have discontinuities and so are expected, in general, to be regular only outside of a set of measure zero. Giusti and Miranda, Giaquinta and Giusti, Ivert and Duzaar and Grotowski methods are presented concerning the proof of such partial regularity results, and optimal regularity is discussed. Finally, Mingione approach is presented about the size of the singular set on which discontinuities may occur. The notes are intended for graduate and postgraduate students with a solid background in functional analysis and some familiarity with partial differential equations.