Nonlinear Elliptic and Parabolic Problems: A Special Tribute to the Work of Herbert Amann

530 Pages·2005·3.857 MB·English

by  H. Abels (auth.), Haim Brezis, Michel Chipot, Joachim Escher (eds.)

Progress in Nonlinear Differential Equations
and Their Applications
Volume 66
Editor
Haim Brezis
Université Pierre et Marie Curie
Paris
and
Rutgers University
New Brunswick, N.J.
Editorial Board
Antonio Ambrosetti, Scuola Internazionale Superiore di Studi Avanzati, Trieste
A. Bahri, Rutgers University, New Brunswick
Felix Browder, Rutgers University, New Brunswick
Luis Cafarelli, Institute for Advanced Study, Princeton
Lawrence C. Evans, University of California, Berkeley
Mariano Giaquinta, University of Pisa
David Kinderlehrer, Carnegie-Mellon University, Pittsburgh
Sergiu Klainerman, Princeton University
Robert Kohn, New York University
P.L. Lions, University of Paris IX
Jean Mahwin, Université Catholique de Louvain
Louis Nirenberg, New York University
Lambertus Peletier, University of Leiden
Paul Rabinowitz, University of Wisconsin, Madison
John Toland, University of Bath
Nonlinear
Elliptic
and Parabolic
Problems
Herbert Amann
A Special Tribute to the Work of
Michel Chipot
Joachim Escher
Editors
V
nda Valent
e
Giorgi
o Vergara Caffarelli
Editors
Birkhäuser
Basel Boston Berlin
(cid:404) (cid:404)
Editors:
Michel Chipot Joachim Escher
Universität Zürich Institute of Applied Mathematics
Angewandte Mathematik University of Hannover
Winterthurerstr. 190 Welfengarten 1
8057 Zürich 30167 Hannover
Switzerland Germany
e-mail: [email protected] e-mail: [email protected]
Carlo Sbordone
Dipartimento di Matematica e Applicazioni
Università di Napoli “Federico II”
Via Cintia
80126 Napoli, Italy
[email protected]
Itai Shafrir
Department of Mathematics
Technion – Israel Institute of Technology
32000 Haifa, Israel
[email protected]
Vanda Valente
CNR-IAC
Viale del Policlinico, 137
00161 Roma, Italy
[email protected]
Giorgio Vergara Caffarelli
Dipartimento di metodi e modelli
matematici per le Scienze Aplicate
Università di Roma “La Sapienza”
Via A. Scarpa 16
00161 Roma, Italy
[email protected]
2000 Mathematics Subject Classification 35Jxx, 35Kxx, 37Gxx, 76xx
A CIP catalogue record for this book is available from the Library of Congress,
Washington D.C., USA
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Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie;
detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>.
ISBN 3-7643-7266-4 Birkhäuser Verlag, Basel – Boston – Berlin
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© 2005 Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland
Part of Springer Science+Business Media
Printed on acid-free paper produced of chlorine-free pulp. TCF ∞
Printed in Germany
ISBN 10: 3-7643-7266-4 e-ISBN 3-7643-7385-7
ISBN 13: 978-3-7643-7266-8
9 8 7 6 5 4 3 2 1 www.birkhauser.ch
Contents
Preface .................................................................. ix
H. Abels
Bounded Imaginary Powers and H∞-Calculus of
the Stokes Operator in Unbounded Domains ........................ 1
M. Borsuk and A. Zawadzka
Exact Estimates of Solutions to the Robin Boundary Value Problem
for Elliptic Non-divergent Second-order Equations in a Neighborhood
of the Boundary Conical Point ...................................... 17
D. Bothe, J. Pru¨ss and G. Simonett
Well-posedness of a Two-phase Flow with Soluble Surfactant ........ 37
J. Brasche and M. Demuth
Resolvent Differences for General Obstacles ......................... 63
J.I. D´ıaz
Special Finite Time Extinction in Nonlinear Evolution
Systems: Dynamic Boundary Conditions and
Coulomb Friction Type Problems ................................... 71
M. Duelli and L. Weis
Spectral Projections, Riesz Transforms and
H∞-calculus for Bisectorial Operators .............................. 99
R. Farwig, G.P. Galdi and H. Sohr
Very Weak Solutions of Stationary and Instationary
Navier-Stokes Equations with Nonhomogeneous Data ............... 113
M. Fila, J.J.L. Vela´zquez and M. Winkler
Grow-up on the Boundary for a Semilinear Parabolic Problem ...... 137
H. Gajewski and I.V. Skrypnik
Existence and Uniqueness Results for Reaction-diffusion Processes
of Electrically Charged Species ...................................... 151
F. Colombo and D. Guidetti
An Inverse Problem for a Phase-field Model in Sobolev Spaces ...... 189
vi Contents
O. Anza Hafsa and M. Chipot
Numerical Analysis of Microstructures:
The Influence of Incompatibility .................................... 211
J. Herna´ndez, F.J. Mancebo and J.M. Vega
Nonlinear Singular Elliptic Problems:
Recent Results and Open Problems ................................. 227
M. Hieber
The Navier-Stokes Flow in the Exterior of Rotating Obstacles ....... 243
M. Kuˇcera, J. Eisner and L. Recke
A Global Bifurcation Result for Variational Inequalities ............. 253
P.C. Kunstmann
On Elliptic Non-divergence Operators with
Measurable Coefficients ............................................. 265
P. Lauren¸cot and D. Wrzosek
A Chemotaxis Model with Threshold Density and
Degenerate Diffusion ................................................ 273
J. Lo´pez-G´omez and M. Molina-Meyer
In the Blink of an Eye .............................................. 291
J. Lo´pez-G´omez and C. Mora-Corral
Generalized Minimal Cardinal of the λ-slices of the Semi-bounded
Components Arising in Global Bifurcation Theory .................. 329
S.A. Messaoudi
Blow-up of Solutions of a Semilinear Heat Equation with
a Visco-elastic Term ................................................ 351
C.M. Murea and G. Hentschel
Finite Element Methods for Investigating the Moving
Boundary Problem in BiologicalDevelopment ....................... 357
J. Naumann
Existence of Weak Solutions to the Equations of Stationary
Motion of Heat-conducting Incompressible Viscous Fluids ........... 373
P. Pola´ˇcik and P. Quittner
Liouville Type Theorems and Complete Blow-up for
Indefinite Superlinear Parabolic Equations .......................... 391
D. Praˇza´k
On Reducing the 2d Navier-Stokes Equations to
a System of Delayed ODEs .......................................... 403
J. Rehberg
Quasilinear Parabolic Equations in Lp .............................. 413
Contents vii
A. Rodr´ıguez-Bernal
Parabolic Equations in Locally Uniform Spaces ..................... 421
B. Scarpellini
Bifurcation of Traveling Waves Related to the B´enardEquations
with an Exterior Force .............................................. 433
H.-J. Schmeißer and W. Sickel
Vector-valued Sobolev Spaces and
Gagliardo-NirenbergInequalities .................................... 463
P. Souplet
The Influence of Gradient Perturbations on Blow-up Asymptotics in
Semilinear Parabolic Problems: A Survey ........................... 473
Kenichiro Umezu
Non-existence of Positive Solutions for Diffusive Logistic Equations
with Nonlinear Boundary Conditions ................................ 497
A. Rodr´ıguez-Bernal and A. Vidal-Lo´pez
Extremal Equilibria and Asymptotic Behavior of
Parabolic Nonlinear Reaction-diffusion Equations ................... 509
C. Walker
A Remark on Continuous Coagulation-FragmentationEquations
with Unbounded Diffusion Coefficients .............................. 517
P. Weidemaier
On L -Estimates of Optimal Type for the Parabolic Oblique
p
Derivative Problem with VMO-Coefficients – A Refined Version ..... 529
Preface
Celebrating the work of a renowned mathematician, it is our pleasure to present
this volume containing the proceedings of the conference “Nonlinear Elliptic and
Parabolic Problems: A Special Tribute to the Work of Herbert Amann”, held in
Zurich, June 28–30, 2004.
Herbert Amann had a significant and decisive impact in developing Non-
linear Analysis and one goal of this conference was to reflect his broad scientific
interest. It is our hope that this collection of papers gives the reader some idea
of the subjects in which Herbert Amann had and still has a deep influence. Of
particular importance are fluid dynamics, reaction-diffusion systems, bifurcation
theory,maximalregularity,evolutionequations,andthetheoryoffunctionspaces.
The organizers thank the following institutions for provided support for the
conference:
• Swiss National Foundation
• Zu¨rcher Hochschulstiftung
• Zu¨rcher Universit¨atsverein
• Mathematisch-naturwissenschaftliche Fakulta¨t (MNF).
Finally,itisourpleasuretothankallcontributors,referees,andBirkha¨userVerlag,
particularly T. Hempfling for their help and cooperation in making possible this
volume.
The Editors
ProgressinNonlinearDifferentialEquations
andTheirApplications,Vol.64,1–15
(cid:1)c 2005Birkh¨auserVerlagBasel/Switzerland
H
Bounded Imaginary Powers and -Calculus
∞
of the Stokes Operator in Unbounded Domains
Helmut Abels
Abstract. In the present contribution we study the Stokes operator Aq =
−Pq∆ on Lqσ(Ω), 1 < q < ∞, where Ω is a suitable bounded or unbounded
domain in Rn, n ≥ 2, with C1,1-boundary. We present some conditions on
Ω and the related function spaces and basic equations which guarantee that
c+Aq for suitable c ∈ R is of positive type and admits a bounded H∞-
calculus. This implies the existence of bounded imaginary powers of c+Aq.
Most domains studied in the theory of Navier-Stokes like, e.g., bounded, ex-
terior, and aperture domains as well as asymptotically flat layers satisfy the
conditions.Theproofisdonebyconstructinganapproximateresolventbased
ontheresultsof[3],whichwereobtainedbyapplyingthecalculusofpseudo-
differential boundary value problems. Finally, the result is used to proof the
existenceofaboundedH∞-calculusoftheStokesoperatorAq onanaperture
domain.
MathematicsSubjectClassification(2000).35Q30,76D07,47A60,47F05.
Keywords. Stokes equations, exterior domains, bounded imaginary powers,
H∞-calculus, aperture domain.
1. Introduction
Inthis articleweconsiderthe StokesoperatorA =−P ∆onLq(Ω)withdomain
q q σ
D(A )={f ∈W2(Ω)n :f| =0}∩Lq(Ω)
q q ∂Ω σ
where P : Lq(Ω)n →Lq(Ω) denotes the Helmholtz projection,
q σ
Lq(Ω):=C∞ (Ω)(cid:2).(cid:2)q, C∞ (Ω):={u∈C∞(Ω)n :divu=0}, andΩ⊆Rn, n≥2,
σ 0,σ 0,σ 0
is a domain specified in Assumption 1.1 below. Properties of the Stokes operator
areimportant for the associatedinstationaryStokesand Navier-Stokesequations.
Since the latter equations arise in mathematical fluid mechanics, many different
kinds of bounded and unbounded domains are of interest and have been studied.
2 H. Abels
The purpose of the present contribution is to present some conditions on Ω
and the related function spaces which guarantee that c+A for suitable c∈R is
q
of positive type and admits a bounded H∞-calculus w.r.t. δ ∈(0,π). Here c+Aq
is of positive type w.r.t. δ if and only if Σ ∪{0}⊆ρ(−c−A ) and
δ q
C
(cid:8)(λ+c+Aq)−1(cid:8)L(Lqσ(Ω)) ≤ |λq,|δ, λ∈Σδ, (1.1)
where Σ := {z ∈ C\{0} : |argz| < δ}. – Note that, if δ > π, (1.1) implies
δ 2
that −c−A generates a bounded, strongly continuous, analytic semi-group. –
q
Moreover, A := c+Aq is said to admit a bounded H∞-calculus w.r.t. δ if and
only if
(cid:1)
1
h(A):= h(−λ)(λ+A)−1dλ (1.2)
2πi
Γ
is a bounded operator satisfying
(cid:8)h(A)(cid:8)L(Lqσ(Ω)) ≤Cq,δ(cid:8)h(cid:8)∞ for all h∈H∞(δ), (1.3)
where H∞(δ) denotes the Banach algebra of all bounded holomorphic functions
h: Σπ−δ → C, cf. McIntosh [25], and Γ is the negatively orientated boundary of
Σδ. We note that in order to prove (1.3) for all h∈H∞(δ) it is sufficient to show
the estimate for h∈H(δ), which consists of all h∈H∞(δ) such that
|z|s
|h(z)|≤C1+|z|2s for all z ∈Σπ−δ
for some s>0, cf. [8, Lemma 2.1]. For h∈H(δ) the integral (1.2) is well defined
as a Bochner integral.
The property of admitting a bounded H∞-calculus is a generalization of
possessingboundedimaginarypowerssincehy(z)=ziy ∈H∞(δ)forallδ ∈(0,π),
which has many important consequences. In particular, (1.3) for δ > π yields the
2
maximal regularity of −c−A by the result of Dore and Venni [13].
q
The resolventestimate (1.1) with arbitrary δ ∈(0,π), 1<q <∞ and c=0,
has been proved for various kinds of domains Ω ⊂ Rn, n ≥ 2, cf. Giga [19] for
boundeddomains,BorchersandSohr[9]andBorchersandVarnhorn[9]forexterior
domains, Farwig and Sohr [16] for aperture domains, Abels and Wiegner [6] for
an infinite layer Ω = Rn−1×(−1,1), and Abels [2] for asymptotically flat layers.
Moreover, see Farwig and Sohr [15] for a general treatment and Desch, Hieber,
and Pru¨ss [12] for the case of a half-space Rn and Rn, where also the borderline
+
cases q =1,∞ have been studied.
ThefactthatA possessesboundedimaginarypowersandadmitsabounded
q
H∞-calculuswasprovedbyGiga[20],GigaandSohr[21],andNollandSaal[27]for
boundeddomainsandforexteriordomainsinRn,n≥3,byGigaandSohr[22]for
thehalf-spaceRn,seealso[12],andbyAbels[5,4,3]fortwo-dimensionalexterior
+
domains, an infinite layer, and asymptotically flat layers.