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This volume deals with the regularity theory for elliptic systems. We may find the origin of such a theory in two of the problems posed by David Hilbert in his celebrated lecture delivered during the International Congress of Mathematicians in 1900 in Paris: 19th problem: Are the solutions to regular problems in the Calculus of Variations always necessarily analytic? 20th problem: does any variational problem have a solution, provided that certain assumptions regarding the given boundary conditions are satisfied, and provided that the notion of a solution is suitably extended? During the last century these two problems have generated a great deal of work, usually referred to as regularity theory, which makes this topic quite relevant in many fields and still very active for research. However, the purpose of this volume, addressed mainly to students, is much more limited. We aim to illustrate only some of the basic ideas and techniques introduced in this context, confining ourselves to important but simple situations and refraining from completeness. In fact some relevant topics are omitted. Topics include: harmonic functions, direct methods, Hilbert space methods and Sobolev spaces, energy estimates, Schauder and L^ptheory both with and without potential theory, including the CalderonZygmund theorem, Harnack's and De GiorgiMoserNash theorems in the scalar case and partial regularity theorems in the vector valued case energy minimizing harmonic maps and minimal graphs in codimension 1 and greater than 1. In this second deeply revised edition we also included the regularity of 2dimensional weakly harmonic maps, the partial regularity ofstationeryharmonic maps, and their connections with the case p=1 of the L^p theory, including the celebrated results of Wente and of CoifmanLionsMeyerSemmes.
About Mariano Giaquinta
Mariano Giaquinta is an Italian mathematician mainly known for his contributions to the fields of calculus of variations, regularity theory of partial differential equation. He is currently professor of mathematics at the Scuola Normale Superiore di Pisa [1] [2] and he is the director of De Giorgi center at Pisa. Luca Martinazzi is professor of mathematics at Rutgers University, Piscataway, USA.
Anarantrius
25 May 2016Michael: Adobe hasn't listed a max storage or max number of documents. Maybe that's still part of what makes this a "tech preview" rather than a finished product.
REPLYJuan
5 January 2016thank you for feed backBB
REPLYTerrazas
27 July 2016Thank you so much. Very thorough and appreciated. I watched the entire video.
REPLYMax
3 May 2016Much to think about here. Thanks for the thoughtful article.
REPLYNidia
10 July 2016Thanks Mark! If you found some of our discussions helpful, please do click on the social sharing buttons at the top or on the left side of our pages to share the knowledge. You never know who else you can also help along the way!
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