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^p-theory both with and without potential theory, including the Calderon-Zygmund theorem, Harnack’s and De Giorgi-Moser-Nash 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 2-dimensional weakly harmonic maps, the partial regularity of stationary harmonic maps, and their connections with the case p=1 of the L^p theory, including the celebrated results of Wente and of Coifman-Lions-Meyer-Semmes.
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^p-theory both with and without potential theory, including the Calderon-Zygmund theorem, Harnack’s and De Giorgi-Moser-Nash 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 2-dimensional weakly harmonic maps, the partial regularity of stationary harmonic maps, and their connections with the case p=1 of the L^p theory, including the celebrated results of Wente and of Coifman-Lions-Meyer-Semmes.
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Mục lục
1 Harmonic functions.- 2 Direct methods.- 3 Hilbert space methods.- 4 L2-regularity: the Caccioppoli inequality.- 5 Schauder estimates.- 6 Some real analysis.- 7 Lp-theory.- 8 The regularity problem in the scalar case.- 9 Partial regularity in the vector-valued case.- 10 Harmonic maps.- 11 A survey of minimal graphs.Giới thiệu về tác giả
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 the University of Basel, Switzerland.Mua cuốn sách điện tử này và nhận thêm 1 cuốn MIỄN PHÍ!
Ngôn ngữ Anh ● định dạng PDF ● Trang 370 ● ISBN 9788876424434 ● Nhà xuất bản Edizioni della Normale ● Thành phố Pisa ● Quốc gia CH ● Được phát hành 2013 ● Phiên bản 2 ● Có thể tải xuống 24 tháng ● Tiền tệ EUR ● TÔI 2721726 ● Sao chép bảo vệ Adobe DRM
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