By Mariano Giaquinta,Luca Martinazzi
19th challenge: Are the options to normal difficulties within the Calculus of diversifications consistently inevitably analytic?
20th challenge: does any variational challenge have an answer, only if convinced assumptions concerning the given boundary stipulations are chuffed, and only if the idea of an answer is certainly extended?
During the final century those difficulties have generated loads of paintings, frequently often called regularity thought, which makes this subject really suitable in lots of fields and nonetheless very energetic for examine.
However, the aim of this quantity, addressed typically to scholars, is far extra constrained. We objective to demonstrate just some of the fundamental principles and methods brought during this context, confining ourselves to big yet easy occasions and refraining from completeness. actually a few proper themes are omitted.
Topics comprise: harmonic capabilities, direct tools, Hilbert area equipment and Sobolev areas, power estimates, Schauder and L^p-theory either with and with out capability thought, 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 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 idea, together with the prestigious result of Wente and of Coifman-Lions-Meyer-Semmes.
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Additional 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