By Mariano Giaquinta,Luca Martinazzi
19th challenge: Are the recommendations to common difficulties within the Calculus of diversifications constantly inevitably analytic?
20th challenge: does any variational challenge have an answer, only if yes assumptions concerning the given boundary stipulations are chuffed, and only if the suggestion of an answer is certainly extended?
During the final century those difficulties have generated loads of paintings, frequently known as regularity conception, which makes this subject particularly appropriate in lots of fields and nonetheless very energetic for examine.
However, the aim of this quantity, addressed ordinarily to scholars, is way extra restricted. We objective to demonstrate just some of the fundamental principles and methods brought during this context, confining ourselves to big yet uncomplicated events and refraining from completeness. in truth a few correct issues are omitted.
Topics contain: harmonic features, direct equipment, 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 bigger than 1.
In this moment deeply revised version we additionally integrated 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 resources 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