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    Polyharmonic Boundary Value Problems: Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains (Lecture Notes in Mathematics) (Paperback) By (author) Filippo Gazzola, By (author) Hans-Christoph Grunau, By (author) Guido Sweers

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    DescriptionLinear elliptic equations arise in several models describing various phenomena in the applied sciences, the most famous being the second order stationary heat eq- tion or,equivalently,the membraneequation. Forthis intensivelywell-studiedlinear problem there are two main lines of results. The ?rst line consists of existence and regularity results. Usually the solution exists and "gains two orders of differen- ation" with respect to the source term. The second line contains comparison type results, namely the property that a positive source term implies that the solution is positive under suitable side constraints such as homogeneous Dirichlet bou- ary conditions. This property is often also called positivity preserving or, simply, maximum principle. These kinds of results hold for general second order elliptic problems, see the books by Gilbarg-Trudinger [198] and Protter-Weinberger [347]. For linear higher order elliptic problems the existence and regularitytype results - main, as one may say, in their full generality whereas comparison type results may fail. Here and in the sequel "higher order" means order at least four. Most interesting models, however, are nonlinear. By now, the theory of second order elliptic problems is quite well developed for semilinear, quasilinear and even for some fully nonlinear problems. If one looks closely at the tools being used in the proofs, then one ?nds that many results bene?t in some way from the positivity preserving property. Techniques based on Harnack's inequality, De Giorgi-Nash- Moser's iteration, viscosity solutions etc.

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  • Full bibliographic data for Polyharmonic Boundary Value Problems

    Polyharmonic Boundary Value Problems
    Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains
    Authors and contributors
    By (author) Filippo Gazzola, By (author) Hans-Christoph Grunau, By (author) Guido Sweers
    Physical properties
    Format: Paperback
    Number of pages: 441
    Width: 156 mm
    Height: 236 mm
    Thickness: 28 mm
    Weight: 762 g
    ISBN 13: 9783642122446
    ISBN 10: 3642122442

    BIC E4L: MAT
    Nielsen BookScan Product Class 3: S7.8
    B&T Book Type: NF
    LC subject heading:
    B&T Merchandise Category: TXT
    LC subject heading:
    B&T General Subject: 710
    B&T Modifier: Region of Publication: 04
    B&T Modifier: Academic Level: 02
    Ingram Subject Code: MA
    BIC subject category V2: TGB
    Warengruppen-Systematik des deutschen Buchhandels: 16220
    BISAC V2.8: TEC009070
    DC22: 515/.35, 515.35
    LC subject heading:
    LC classification: QA808.2
    BIC subject category V2: PBKF
    BISAC V2.8: MAT012030
    LC subject heading:
    Libri: MECH2000
    BISAC V2.8: MAT000000
    BIC subject category V2: PBMP
    BISAC V2.8: MAT037000
    Libri: MATH3500, ANAL8025, FUNK5300
    LC subject heading:
    Libri: DIFF3000, GEOM5012
    LC classification: QA379 .G39 2010
    LC subject heading: ,
    LC classification: QA1-939, QA319-329.9, TA405-409.3, QA641-670
    Thema V1.0: TGB, PBMP, PBKF
    Edition statement
    2010 ed.
    Illustrations note
    18 black & white illustrations, biography
    Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
    Imprint name
    Springer-Verlag Berlin and Heidelberg GmbH & Co. K
    Publication date
    30 June 2010
    Publication City/Country
    Review quote
    From the reviews: "This is an excellent book, full of well-explained ideas and techniques on the subject, and can be used as a textbook in an advanced course dealing with higher-order elliptic problems. The proofs of almost all of the theorems directly related to the higher-order elliptic problems are complete and well written. In general, the book itself is written in a very clear, pleasurable style, including a wealth of useful diagrams and figures, some of them in color." (Rodney Josue Biezuner, Mathematical Reviews, Issue 2011 h) "The main tasks of the present Lecture Notes in Mathematics volume are nonlinear problems and positivity statements for higher order elliptic equations involving polyharmonic operators. In particular the biharmonic operator and semilinear operators related to it are investigated. ... That the authors are experienced researchers on the topic of the volume becomes evident from the excellent, clear and well motivated presentation. ... Whoever is interested in an exciting subject of the modern theory of higher order elliptic equations is recommended to study this exposition." (Heinrich Begehr, Zentralblatt MATH, Vol. 1239, 2012)
    Back cover copy
    This 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. Underlying models and, in particular, the role of different boundary conditions are explained in detail. As for linear problems, after a brief summary of the existence theory and Lp and Schauder estimates, the focus is on positivity or - since, in contrast to second order equations, a general form of a comparison principle does not exist - on near positivity. The required kernel estimates are also presented in detail. As for nonlinear problems, several techniques well-known from second order equations cannot be utilized and have to be replaced by new and different methods. Subcritical, critical and supercritical nonlinearities are discussed and various existence and nonexistence results are proved. The interplay with the positivity topic from the rst part is emphasized and, moreover, a far-reaching Gidas-Ni-Nirenberg-type symmetry result is included. Finally, some recent progress on the Dirichlet problem for Willmore surfaces under symmetry assumptions is discussed."
    Table of contents
    Models of Higher Order.- Linear Problems.- Eigenvalue Problems.- Kernel Estimates.- Positivity and Lower Order Perturbations.- Dominance of Positivity in Linear Equations.- Semilinear Problems.- Willmore Surfaces of Revolution.