Isoperimetric Inequalities for Laplace and Steklov Problems on Riemannian Manifolds PDF Download
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Author: Luc Pétiard
Publisher:
ISBN:
Category :
Languages : en
Pages : 65
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Book Description
Author: Luc Pétiard
Publisher:
ISBN:
Category :
Languages : en
Pages : 65
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Book Description
Author: Mikhail Karpukhin
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Book Description
"Isoperimetric inequalities for eigenvalues of geometric operators have received a lot of attention in recent years. Such inequalities are of particular interest for the eigenvalues of Laplace and Dirichlet-to-Neumann operators on Riemannian manifolds, where they exhibit a surprising connection to the theory of minimal submanifolds. This bridge between analysis and geometry provides a path to solving problems from both fields. In the present manuscript we apply geometric methods to prove several new isoperimetric inequalities. In particular, we obtain an isoperimetric inequality for the first Laplace eigenvalue on non-orientable surfaces, improving upon results of P. Li and S.-T. Yau. We prove an isoperimetric inequality for all Steklov eigenvalues on orientable surfaces, improving upon results of A. Girouard and I. Polterovich. We also provide a high-dimensional analog of the latter inequality by considering Dirichlet-to-Neumann operators on differential forms. Finally, jointly with N. Nadirashvili, A. Penskoi and I. Polterovich we establish a sharp inequality for all Laplace eigenvalues on a two-dimensional sphere settling the conjecture of Nadirashvili proposed in 2002." --
Author: Isaac Chavel
Publisher: Cambridge University Press
ISBN: 9780521802673
Category : Mathematics
Languages : en
Pages : 292
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Book Description
This advanced introduction emphasizes the variety of ideas, techniques, and applications of the subject.
Author: Gabriel-Eduard Vîlcu
Publisher: MDPI
ISBN: 303650298X
Category : Mathematics
Languages : en
Pages : 208
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Book Description
This book presents the recent developments in the field of geometric inequalities and their applications. The volume covers a vast range of topics, such as complex geometry, contact geometry, statistical manifolds, Riemannian submanifolds, optimization theory, topology of manifolds, log-concave functions, Obata differential equation, Chen invariants, Einstein spaces, warped products, solitons, isoperimetric problem, Erdös–Mordell inequality, Barrow’s inequality, Simpson inequality, Chen inequalities, and q-integral inequalities. By exposing new concepts, techniques and ideas, this book will certainly stimulate further research in the field.
Author: Catherine Bandle
Publisher: Pitman Publishing
ISBN:
Category : Mathematics
Languages : en
Pages : 248
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Book Description
Author: Michael Levitin
Publisher: American Mathematical Society
ISBN: 1470475251
Category : Mathematics
Languages : en
Pages : 346
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Book Description
It is remarkable that various distinct physical phenomena, such as wave propagation, heat diffusion, electron movement in quantum mechanics, oscillations of fluid in a container, can be described using the same differential operator, the Laplacian. Spectral data (i.e., eigenvalues and eigenfunctions) of the Laplacian depend in a subtle way on the geometry of the underlying object, e.g., a Euclidean domain or a Riemannian manifold, on which the operator is defined. This dependence, or, rather, the interplay between the geometry and the spectrum, is the main subject of spectral geometry. Its roots can be traced to Ernst Chladni's experiments with vibrating plates, Lord Rayleigh's theory of sound, and Mark Kac's celebrated question “Can one hear the shape of a drum?” In the second half of the twentieth century spectral geometry emerged as a separate branch of geometric analysis. Nowadays it is a rapidly developing area of mathematics, with close connections to other fields, such as differential geometry, mathematical physics, partial differential equations, number theory, dynamical systems, and numerical analysis. This book can be used for a graduate or an advanced undergraduate course on spectral geometry, starting from the basics but at the same time covering some of the exciting recent developments which can be explained without too many prerequisites.
Author: Pierre H. Berard
Publisher: Springer
ISBN: 3540409580
Category : Mathematics
Languages : en
Pages : 284
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Book Description
Author: Menahem Schiffer
Publisher: Courier Corporation
ISBN: 0486795438
Category : Mathematics
Languages : en
Pages : 465
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Book Description
This advanced monograph on finite Riemann surfaces, based on the authors' 1949–50 lectures at Princeton University, remains a fundamental book for graduate students. The Bulletin of the American Mathematical Society hailed the self-contained treatment as the source of "a plethora of ideas, each interesting in its own right," noting that "the patient reader will be richly rewarded." Suitable for graduate-level courses, the text begins with three chapters that offer a development of the classical theory along historical lines, examining geometrical and physical considerations, existence theorems for finite Riemann surfaces, and relations between differentials. Subsequent chapters explore bilinear differentials, surfaces imbedded in a given surface, integral operators, and variations of surfaces and of their functionals. The book concludes with a look at applications of the variational method and remarks on generalization to higher dimensional Kahler manifolds.
Author: Antoine Henrot
Publisher: De Gruyter Open
ISBN: 9783110550856
Category :
Languages : en
Pages : 474
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Book Description
"Shape optimization and spectral theory" is a survey book aiming to give an overview of recent results in spectral geometry and its links with shape optimization. It covers most of the issues which are important for people working in PDE and differential geometry interested in sharp inequalities and qualitative behaviour for eigenvalues of the Laplacian with different kind of boundary conditions (Dirichlet, Robin and Steklov). This includes: existence of optimal shapes, their regularity, the case of special domains like triangles, isospectrality, quantitative form of the isoperimetric inequalities, optimal partitions, universal inequalities and numerical results. Much progress has been made in these extremum problems during the last ten years and this edited volume presents a valuable update to a wide community interested in these topics. List of contributors Antunes Pedro R.S., Ashbaugh Mark, Bonnaillie-Noel Virginie, Brasco Lorenzo, Bucur Dorin, Buttazzo Giuseppe, De Philippis Guido, Freitas Pedro, Girouard Alexandre, Helffer Bernard, Kennedy James, Lamboley Jimmy, Laugesen Richard S., Oudet Edouard, Pierre Michel, Polterovich Iosif, Siudeja Bartlomiej A., Velichkov Bozhidar
Author: Filippo Gazzola
Publisher: Springer
ISBN: 3642122450
Category : Mathematics
Languages : en
Pages : 444
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Book Description
This accessible monograph covers higher order linear and nonlinear elliptic boundary value problems in bounded domains, mainly with the biharmonic or poly-harmonic operator as leading principal part. It provides rapid access to recent results and references.
