By Mariano Giaquinta,Luca Martinazzi

This quantity offers with the regularity concept for elliptic structures. We could locate the beginning of this type of thought in of the issues posed by means of David Hilbert in his celebrated lecture brought through the overseas Congress of Mathematicians in 1900 in Paris:
19th challenge: Are the ideas to commonplace difficulties within the Calculus of adaptations constantly unavoidably analytic?
20th challenge: does any variational challenge have an answer, only if yes assumptions concerning the given boundary stipulations are happy, and only if the suggestion of an answer is certainly extended?
During the final century those difficulties have generated loads of paintings, often known as regularity concept, which makes this subject particularly correct in lots of fields and nonetheless very lively for study.
However, the aim of this quantity, addressed ordinarily to scholars, is way extra restricted. We goal to demonstrate just some of the fundamental principles and strategies brought during this context, confining ourselves to special yet easy occasions and refraining from completeness. actually a few suitable themes are omitted.
Topics comprise: harmonic features, direct tools, Hilbert area tools and Sobolev areas, strength estimates, Schauder and L^p-theory either with and with no strength conception, together with the Calderon-Zygmund theorem, Harnack's and De Giorgi-Moser-Nash theorems within the scalar case and partial regularity theorems within the vector valued case; strength minimizing harmonic maps and minimum graphs in codimension 1 and bigger than 1.
In this moment deeply revised variation we additionally incorporated the regularity of 2-dimensional weakly harmonic maps, the partial regularity of desk bound harmonic maps, and their connections with the case p=1 of the L^p thought, together with the distinguished result of Wente and of Coifman-Lions-Meyer-Semmes.

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