Polyharmonic Boundary Value Problems Positivity Preserving and Nonli
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 bounda
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1991
Filippo Gazzola · Hans-Christoph Grunau Guido Sweers
Polyharmonic Boundary Value Problems Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains
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Filippo Gazzola
Guido Sweers
Dipartimento di Matematica Politecnico di Milano Piazza Leonardo da Vinci 32 20133 Milano, Italy [email protected]
Mathematisches Institut Universit¨at zu K¨oln Weyertal 86-90 50931 K¨oln, Germany [email protected]
Hans-Christoph Grunau Institut f¨ur Analysis und Numerik Otto von Guericke-Universit¨at Postfach 4120 39016 Magdeburg, Germany [email protected]
ISBN: 978-3-642-12244-6 e-ISBN: 978-3-642-12245-3 DOI: 10.1007/978-3-642-12245-3 Springer Heidelberg Dordrecht London New York Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2010927754 Mathematics Subject Classification (2000): 35J40, 35J66, 35J91, 35B50, 35B45, 35J35, 35J62, 46E35, 53C42, 74K20 c Springer-Verlag Berlin Heidelberg 2010 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, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: SPi Publisher Services Printed on acid-free paper springer.com
Dedicated to our wives Chiara, Brigitte and Barbara.
The cover figure displays the solution of Δ 2 u = f in a rectangle with homogeneous Dirichlet boundary condition for a nonnegative function f with its support concentrated near a point on the left hand side. The dark part shows the region where u < 0.
Preface
Linear elliptic equations arise in several models describing various phenomena in the applied sciences, the most famous being the second order stationary heat equation or, equivalently, the membrane equation. For this intensively well-studied linear problem there are two main lines of results. The first line consists of existence and regularity results. Usually the solution exists and “gains two orders of differentiation” 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 boundary conditions. This property is often also called positivity preserving or, simply, maximum principle. These kinds of resu
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