By Mariano Giaquinta,Luca Martinazzi
19th challenge: Are the ideas to average difficulties within the Calculus of diversifications constantly unavoidably analytic?
20th challenge: does any variational challenge have an answer, only if definite assumptions concerning the given boundary stipulations are chuffed, and only if the proposal of an answer is definitely extended?
During the final century those difficulties have generated loads of paintings, frequently often called regularity idea, which makes this subject fairly proper in lots of fields and nonetheless very energetic for learn.
However, the aim of this quantity, addressed generally to scholars, is far extra constrained. We objective to demonstrate just some of the fundamental rules and strategies brought during this context, confining ourselves to big yet uncomplicated events and refraining from completeness. in truth a few suitable themes are omitted.
Topics contain: harmonic services, direct tools, Hilbert area tools and Sobolev areas, strength estimates, Schauder and L^p-theory either with and with out capability idea, 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 larger 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 concept, together with the distinguished result of Wente and of Coifman-Lions-Meyer-Semmes.
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Extra info for An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs (Publications of the Scuola Normale Superiore)
An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs (Publications of the Scuola Normale Superiore) by Mariano Giaquinta,Luca Martinazzi