Estimates for Differential Operators in Half-space (EMS Tracts in Mathematics)

264 Pages·2019·1.174 MB·English

by  Igor W. Gel'man, Vladimir G. maz'ya

Tracts in Mathematics 31 VI Tracts in Mathematics 31
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Igor V. Gel’man a
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Vladimir G. Maz’ya ’
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Estimates for Differential Operators
Vladimir G. Maz’ya
in Half-space
Inequalities for differential operators play a fundamental role in the modern iE Estimates for
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theory of partial differential equations. Among the numerous app lications of s
such inequalities are existence and uniqueness theorems, error estimates for Ht
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numerical approximations of solutions and for residual terms in a symptotic am
Differential Operators
formulas, as well as results on the structure of the spectrum. The inequalities lfa
cover a wide range of differential operators, boundary conditions and norms -t
of the corresponding function spaces. se
ps in Half-space
The book focuses on estimates up to the boundary of a domain. It contains a f
a great variety of inequalities for differential and pseudodifferent ial co
operators with constant coefficients. Results of final character a re obtained, er

without any restrictions on the type of differential operators. Algebraic D
necessary and sufficient conditions for the validity of the corresponding a i
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priori estimates are presented. General criteria are systematically applied
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to particular types of operators found in classical equations and systems e
of mathematical physics (such as Lame’s system of static ela sticity theory r
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or the linearized Navier–Stokes system), Cauchy–Riemann’s opera tors,
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Schrödinger operators, among others. The well-known results of Aronszajn,
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Agmon–Douglis–Nirenberg and Schechter fall into the general scheme, and i
sometimes are strengthened. a
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The book will be interesting and useful to a wide audience, including graduate
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students and specialists in the theory of differential equations.
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ISBN 978-3-03719-191-0
www.ems-ph.org
Gelman_Mazya Cover | Font: Nuri_Helvetica Neue | Farben: Pantone 116, Pantone 287 | RB 26 mm
EMS Tracts in Mathematics 31
EMS Tracts in Mathematics
Editorial Board:
Michael Farber (Queen Mary University of London, Great Britain)
Carlos E. Kenig (The University of Chicago, USA)
Michael Röckner (Universität Bielefeld, Germany, and Purdue University, USA)
Vladimir Turaev (Indiana University, Bloomington, USA)
Alexander Varchenko (The University of North Carolina at Chapel Hill, USA)
This series includes advanced texts and monographs covering all fields in pure and applied
mathematics. Tracts will give a reliable introduction and reference to special fields of
current research. The books in the series will in most cases be authored monographs,
although edited volumes may be published if appropriate. They are addressed to graduate
students seeking access to research topics as well as to the experts in the field working
at the frontier of research.
For a complete listing see our homepage at www.ems-ph.org.
13 Laurent Bessières et al., Geometrisation of 3-Manifolds
14 Steffen Börm, Efficient Numerical Methods for Non-local Operators. 2-Matrix
Compression, Algorithms and Analysis
15 Ronald Brown, Philip J. Higgins and Rafael Sivera, Nonabelian Algebraic Topology.
Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids
16 Marek Janicki and Peter Pflug, Separately Analytical Functions
17 Anders Björn and Jana Björn, Nonlinear Potential Theory on Metric Spaces
18 Erich Novak and Henryk Woz´niakowski, Tractability of Multivariate Problems.
Volume III: Standard Information for Operators
19 Bogdan Bojarski, Vladimir Gutlyanskii, Olli Martio and Vladimir Ryazanov, Infinitesimal
Geometry of Quasiconformal and Bi-Lipschitz Mappings in the Plane
20 Hans Triebel, Local Function Spaces, Heat and Navier–Stokes Equations
21 Kaspar Nipp and Daniel Stoffer, Invariant Manifolds in Discrete and Continuous
Dynamical Systems
22 Patrick Dehornoy with François Digne, Eddy Godelle, Daan Kramer and Jean Michel,
Foundations of Garside Theory
23 Augusto C. Ponce, Elliptic PDEs, Measures and Capacities. From the Poisson Equation
to Nonlinear Thomas–Fermi Problems
24 Hans Triebel, Hybrid Function Spaces, Heat and Navier-Stokes Equations
25 Yves Cornulier and Pierre de la Harpe, Metric Geometry of Locally Compact Groups
26 Vincent Guedj and Ahmed Zeriahi, Degenerate Complex Monge–Ampère Equations
27 Nicolas Raymond, Bound States of the Magnetic Schrödinger Operator
28 Antoine Henrot and Michel Pierre, Shape Variation and Optimization. A Geometrical Analysis
29 Alexander Kosyak, Regular, Quasi-regular and Induced Representations of Infinite-
dimensional Groups
30 Vladimir G. Maz’ya, Boundary Behavior of Solutions to Elliptic Equations in General Domains
Igor V. Gel’man
Vladimir G. Maz’ya
Estimates for
Differential Operators
in Half-space
Translated from the German by
Darya Apushkinskaya
Authors:
Igor V. Gel’man
Akko 24103
Israel
Vladimir G. Maz’ya
Department of Mathematics and University of Liverpool, UK
Linköping University
and
581 83 Linköping
Peoples’ Friendship University of Russia
Sweden
(RUDN University), Moscow, Russian Federation
E-mail: [email protected]
Originally published in 1981 by Akademie-Verlag under the title
Abschätzungen für Differentialoperatoren im Halbraum
2010 Mathematical Subject Classification: 35G05, 35S05
Key words: Differential operators with constant coefficients, differential operators in a half-space,
pseudo-differential operators, domination of differential operators, boundary traces, maximal opera-
tor, estimates for Lame system, estimates for Stokes system
ISBN 978-3-03719-191-0
The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and
the detailed bibliographic data are available on the Internet at http://www.helveticat.ch.
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is
concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broad-
casting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use
permission of the copyright owner must be obtained.
© 2019 European Mathematical Society
Contact address:
European Mathematical Society Publishing House
Seminar for Applied Mathematics
ETH-Zentrum SEW A21
CH-8092 Zürich
Switzerland
Email: [email protected]
Homepage: www.ems-ph.org
Typeset using the authors’ TEX files: le-tex publishing services GmbH, Leipzig, Germany
Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany
∞ Printed on acid free paper
9 8 7 6 5 4 3 2 1
Dedicatedto
SolomonGrigorievichMikhlin
Whenaproblemaboutpartialdifferentialoperatorshasbeenfitted
intotheabstracttheory,allthatremainsisusuallytoprovea
suitableinequalityandmuchofournewknowledgeis,infact,
essentiallycontainedinsuchinequalities.Buttheabstracttheoryis
notonlyatool,itisalsoaguidetogeneralandfruitfulproblems.
LarsGa˚rding

Translator’s Preface
Thehistoryofthisbookissomewhatunusual. OriginallyitwaswritteninLeningrad
in Russian, at the end of the seventies. Submitted to the Soviet publishing house
“Nauka”, it was rejected for political reasons and never appeared in Russian. The
manuscriptwassmuggledtoEastGermanybyfriendsoftheauthors,translatedinto
German, and published in 1981 underthe title “Abscha¨tzungenfu¨r Differentialope-
ratorenimHalbraum”byAkademie-VerlagBerlin. In1982,theGermaneditionwas
reprintedby Birkha¨userVerlag. Now, almostfortyyearslater, the English-speaking
readerscanalsobecomefamiliarwiththisinterestingbook.
The book is devoted to a detailed study of numerous inequalities for differen-
tial operators with constant coefficients in a half-space. Although many years have
passed since the book was written, the presented results are not outdated. They
are in definitive form, without any restriction on the type of differential operators.
Moreover, the presented inequalities open perspectives for possible generalizations
to operators in other domains and operators with variable coefficients, as well as to
pseudo-differentialoperators.
D.Apushkinskaya

Authors’ Acknowledgements
We would like to express our deep gratitude to Thomas Hintermann and Sylvia
Fellmann for deciding to publish this book. Also our cordial thanks are to Darya
Apushkinskayaforthetranslationofthisbook.
Thispublicationwaspreparedwiththesupportofthe“RUDNUniversityProgram
5-100”.