J. Funct. Anal., 201(1):298–300, 2003. 246. T. Mengesha. Nonlocal Korn-type characterization of Sobolev vector fields. Commun. Contemp. Math., 14(4):1250028, 28, 2012. 247. B. Merlet. Two remarks on liftings of maps with values into S1.
Sobolev Maps to the Circle will appeal to mathematicians working in various areas, such as nonlinear analysis, PDEs, geometric analysis, minimal surfaces, optimal transport, and topology.
Author: Haïm Brezis
The theory of real-valued Sobolev functions is a classical part of analysis and has a wide range of applications in pure and applied mathematics. By contrast, the study of manifold-valued Sobolev maps is relatively new. The incentive to explore these spaces arose in the last forty years from geometry and physics. This monograph is the first to provide a unified, comprehensive treatment of Sobolev maps to the circle, presenting numerous results obtained by the authors and others. Many surprising connections to other areas of mathematics are explored, including the Monge-Kantorovich theory in optimal transport, items in geometric measure theory, Fourier series, and non-local functionals occurring, for example, as denoising filters in image processing. Numerous digressions provide a glimpse of the theory of sphere-valued Sobolev maps. Each chapter focuses on a single topic and starts with a detailed overview, followed by the most significant results, and rather complete proofs. The “Complements and Open Problems” sections provide short introductions to various subsequent developments or related topics, and suggest newdirections of research. Historical perspectives and a comprehensive list of references close out each chapter. Topics covered include lifting, point and line singularities, minimal connections and minimal surfaces, uniqueness spaces, factorization, density, Dirichlet problems, trace theory, and gap phenomena. Sobolev Maps to the Circle will appeal to mathematicians working in various areas, such as nonlinear analysis, PDEs, geometric analysis, minimal surfaces, optimal transport, and topology. It will also be of interest to physicists working on liquid crystals and the Ginzburg-Landau theory of superconductors.
The lifting problems for Sobolev maps into the circle has been recently thoroughly analyzed by Haim Brezis and his collaborators. In this paper, we wish to address the lifting problem for more general manifolds as well as its extension ...
Author: Henri Berestycki
Publisher: American Mathematical Soc.
In celebration of Haim Brezis's 60th birthday, a conference was held at the Ecole Polytechnique in Paris, with a program testifying to Brezis's wide-ranging influence on nonlinear analysis and partial differential equations. The articles in this volume are primarily from that conference. They present a rare view of the state of the art of many aspects of nonlinear PDEs, as well as describe new directions that are being opened up in this field. The articles, written by mathematicians at the center of current developments, provide somewhat more personal views of the important developments and challenges.
We underline that all the transformations used to reduce to constant coefficients the water waves system (1.3.5) up to smoothing remainders are defined by paradifferential operators, and they are bounded maps acting on Sobolev spaces ...
Author: Massimiliano Berti
The goal of this monograph is to prove that any solution of the Cauchy problem for the capillary-gravity water waves equations, in one space dimension, with periodic, even in space, small and smooth enough initial data, is almost globally defined in time on Sobolev spaces, provided the gravity-capillarity parameters are taken outside an exceptional subset of zero measure. In contrast to the many results known for these equations on the real line, with decaying Cauchy data, one cannot make use of dispersive properties of the linear flow. Instead, a normal forms-based procedure is used, eliminating those contributions to the Sobolev energy that are of lower degree of homogeneity in the solution. Since the water waves equations form a quasi-linear system, the usual normal forms approaches would face the well-known problem of losses of derivatives in the unbounded transformations. To overcome this, after a paralinearization of the capillary-gravity water waves equations, we perform several paradifferential reductions to obtain a diagonal system with constant coefficient symbols, up to smoothing remainders. Then we start with a normal form procedure where the small divisors are compensated by the previous paradifferential regularization. The reversible structure of the water waves equations, and the fact that we seek solutions even in space, guarantees a key cancellation which prevents the growth of the Sobolev norms of the solutions.
Author: Themistocles M. RassiasPublish On: 2018-06-18
Non Linéaire (2018, to appear). https://hal.archives-ouvertes.fr/ hal-01626613 5. H. Brezis, P. Mironescu, Sobolev Maps with Values Into the Circle (Birkhäuser, Basel) (in preparation) 6. H. Brezis, P. Mironescu, Gagliardo-Nirenberg, ...
Author: Themistocles M. Rassias
Current research and applications in nonlinear analysis influenced by Haim Brezis and Louis Nirenberg are presented in this book by leading mathematicians. Each contribution aims to broaden reader’s understanding of theories, methods, and techniques utilized to solve significant problems. Topics include: Sobolev Spaces Maximal monotone operators A theorem of Brezis-Nirenberg Operator-norm convergence of the Trotter product formula Elliptic operators with infinitely many variables Pseudo-and quasiconvexities for nonsmooth function Anisotropic surface measures Eulerian and Lagrangian variables Multiple periodic solutions of Lagrangian systems Porous medium equation Nondiscrete Lassonde-Revalski principle Graduate students and researchers in mathematics, physics, engineering, and economics will find this book a useful reference for new techniques and research areas. Haim Brezis and Louis Nirenberg’s fundamental research in nonlinear functional analysis and nonlinear partial differential equations along with their years of teaching and training students have had a notable impact in the field.
H. Brezis and P. Mironescu, Sobolev Maps to the Circle, Birkhäuser, Basel, to appear. 75. L.E.J. Brouwer, Beweis der Invarianz der Dimensionenzahl, Math. Ann. 70 (1911), 161–165. 76. L.E.J. Brouwer, Ueber Abbildungen von ...
Author: George Dinca
Publisher: Springer Nature
This monograph explores the concept of the Brouwer degree and its continuing impact on the development of important areas of nonlinear analysis. The authors define the degree using an analytical approach proposed by Heinz in 1959 and further developed by Mawhin in 2004, linking it to the Kronecker index and employing the language of differential forms. The chapters are organized so that they can be approached in various ways depending on the interests of the reader. Unifying this structure is the central role the Brouwer degree plays in nonlinear analysis, which is illustrated with existence, surjectivity, and fixed point theorems for nonlinear mappings. Special attention is paid to the computation of the degree, as well as to the wide array of applications, such as linking, differential and partial differential equations, difference equations, variational and hemivariational inequalities, game theory, and mechanics. Each chapter features bibliographic and historical notes, and the final chapter examines the full history. Brouwer Degree will serve as an authoritative reference on the topic and will be of interest to professional mathematicians, researchers, and graduate students.
There are several mapping spaces between manifolds which can be viewed as Hilbert spaces by only considering maps of suitable Sobolev class. For example we can consider the space LM of all H1 maps from the unit circle S1 into a manifold ...
Author: Carmen Hurley &
Publisher: Scientific e-Resources
The idea of "e;ondelettes"e; or "e;wavelets"e; started from the investigation of time-frequency signal analysis, wave engendering, and examining hypothesis. One of the principle explanations behind the disclosure of wavelets and wavelet changes is that the Fourier change analysis does not contain the neighborhood data of signals. So the Fourier change can't be utilized for examining signals in a joint time and frequency area. In 1982, Jean MorIet, in a joint effort with a gathering of French designers, first presented the possibility of wavelets as a group of capacities built by utilizing interpretation and expansion of a solitary capacity, called the mother wavelet, for the analysis of nonstationary signals. Wavelet analysis is an energizing new technique for taking care of troublesome issues in science, material science, and building, with present day applications as various as wave spread, information pressure, picture preparing, design acknowledgment, PC illustrations, the location of air ship and submarines, and change in CAT filters and other restorative picture innovation. Wavelets permit complex data, for example, music, discourse, pictures, and examples to be deteriorated into basic structures, called the central building hinders, at various positions and scales and in this manner remade with high accuracy.
Author: Mikhail Aleksandrovich ShubinPublish On: 2011-02-10
Volume 535, 2011 Sobolev Mapping Properties of the Scattering Transform for the Schrödinger Equation Rostyslav O. Hryniv ... An analogous representation for Schrödinger operators on the circle appears in the work of Kappeler and Topalov ...
Author: Mikhail Aleksandrovich Shubin
Publisher: American Mathematical Soc.
The papers in this volume cover important topics in spectral theory and geometric analysis such as resolutions of smooth group actions, spectral asymptotics, solutions of the Ginzburg-Landau equation, scattering theory, Riemann surfaces of infinite genus and tropical mathematics.
merely in the Sobolev space HR with s > 3/2 ( for a detailed proof of this see ( M2 ) in the periodic case and ( LO ) ... Consider the set H $ ( T , T ) of maps from the circle into itself that are of Sobolev class HR in every chart on T.
... Control and relaxation over the circle, 2000 Robert Rumely, Chi Fong Lau, and Robert Varley, Existence of the sectional capacity, ... 2000 Piotr Hajtasz and Pekka Koskela, Sobolev met Poincaré, 2000 Guy David and Stephen Semmes, ...
Author: Bruce Hughes
Publisher: American Mathematical Soc.
This work formulates and proves a geometric version of the fundamental theorem of algebraic K-theory which relates the K-theory of the Laurent polynomial extension of a ring to the K-theory of the ring. The geometric version relates the higher simple homotopy theory of the product of a finite complex and a circle with that of the complex. By using methods of controlled topology, we also obtain a geometric version of the fundamental theorem of lower algebraic K-theory. The main new innovation is a geometrically defined nil space.