An Introduction to the Regularity Theory for Elliptic by Mariano Giaquinta,Luca Martinazzi

By Mariano Giaquinta,Luca Martinazzi

This quantity offers with the regularity thought for elliptic structures. We could locate the beginning of any such conception in of the issues posed by means of David Hilbert in his celebrated lecture introduced throughout the overseas Congress of Mathematicians in 1900 in Paris:
19th challenge: Are the strategies to ordinary difficulties within the Calculus of diversifications continuously inevitably 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 idea of an answer is definitely extended?
During the final century those difficulties have generated loads of paintings, often known as regularity idea, which makes this subject fairly appropriate in lots of fields and nonetheless very energetic for learn.
However, the aim of this quantity, addressed usually to scholars, is way extra constrained. We target to demonstrate just some of the elemental principles and methods brought during this context, confining ourselves to big yet basic events and refraining from completeness. in truth a few appropriate issues are omitted.
Topics comprise: harmonic features, direct tools, Hilbert area tools and Sobolev areas, power estimates, Schauder and L^p-theory either with and with out power concept, 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; power minimizing harmonic maps and minimum graphs in codimension 1 and larger than 1.
In this moment deeply revised version 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 idea, together with the prestigious result of Wente and of Coifman-Lions-Meyer-Semmes.

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