Table Of ContentProgress in Nonlinear Differential Equations
and Their Applications
Volume 55
Editor
Haim Brezis
Universit 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
RL. Lions, University of Paris IX
Jean Mahwin, Universit Catholique de Louvain
Louis Nirenberg, New York University
Lambertus Peletier, University of Leiden
Paul Rabinowitz, University of Wisconsin, Madison
John Toland, University of Bath
Evolution Equations:
Applications to Physics, Industry,
Life Sciences and Economics
EVEQ2000 Conference in Levico Terme (Trento, Italy),
October 30-November 4, 2000
Mimmo Iannelli
Günter Lumer
Editors
Springer Basel AG
Editors' addresses:
Mimmo Iannelli Günter Lumer
Department of Mathematics Institute of Mathematics and Informatics
University of Trento University of Mons-Hainaut
via Sommarive 14 20, place du Pare
38050 Povo, Trento 7000 Möns
Italy Belgium
e-mail: [email protected] e-mail: [email protected]
2000 Mathematics Subject Classification: 34-xx, 35-xx, 47Dxx, 49Lxx, 58Jxx, 60Jxx,
76Mxx, 76Nxx, 91Bxx, 92Bxx, 92Cxx
A CIP catalogue record for this book is available from the Library of Congress,
Washington D.C., USA
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ISBN 978-3-0348-9433-3 ISBN 978-3-0348-8085-5 (eBook)
DOI 10.1007/978-3-0348-8085-5
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Originally published by Birkhäuser Verlag in 2003
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Table of Contents
Preface ................................................................... Vll
F. Andreu, V. Caselles and 1.M. Mazon
Evolution problems associated to linear growth functionals:
the Dirichlet problem ................................................ 1
/. Antoniou and Z. Suchanecki
Time operators associated to dilations of Markov processes 13
1.-P. Aubin
Boundary-value problems for systems of Hamilton-Jacobi-Bellman
inclusions with constraints ........................................... 25
A. V. Balakrishnan
On a convolution-evolution equation in aeroelasticity 61
A. Britkai and B. Farkas
On the effect of small delays to the stability of feedback systems 83
Z. Brzeiniak, B. Goldys and 1.M.A.M. van Neerven
Mean square continuity of Ornstein-Uhlenbeck processes
in Banach spaces .................................................... 95
P. Clement and 1. pruss
Some remarks on maximal regularity of parabolic problems 101
F. Colombo, M. Giuli and V. Vespri
A semigroup approach to no-arbitrage pricing theory:
constant elasticity variance model and term structure models 113
G. Da Prato
Perturbations of Ornstein-Uhlenbeck operators:
an analytic approach ................................................ 127
R. Denk, M. Hieber and 1. pruss
Towards an L1-theory for vector-valued elliptic
boundary value problems ............................................ 141
1. Escher
On moving boundaries in elliptic-parabolic systems 149
E. Fasanga
Spectral mapping theorems and spectral space-independence 157
A. Favini, G. Ruiz Goldstein, 1.A. Goldstein, E. Obrecht and S. Romanelli
The Laplacian with generalized Wentzell boundary conditions. . . . .. .. 169
VI Table of Contents
S. Flory, F. Neubrander and L. Weis
Consistency and stabilization of rational approximation schemes
for Co-semigroups ................................................... 181
A.A. Gavrilov, S. Nicaise and O.M. Penkin
Poincare's inequality on stratified sets and applications .............. 195
K.P. Hadeler and C. Kuttler
Variational principles for granular matter ............................ 215
M. Iannelli and M. M artcheva
Homogeneous dynamical systems and the age-structured
SIR model with proportionate mixing incidence ...................... 227
V. G. Jakubowski and WM. Ruess
Asymptotic behaviour of solutions of nonlinear
Volterra equations ................................................... 253
M. Kramar, D. Mugnolo and R. Nagel
Semigroups for initial boundary value problems ...................... 275
P. C. K unstmann
Maximal Lp-regularity for second order elliptic operators
with uniformly continuous coefficients on domains ................... 293
M. Langlais and J-M. N aulin
An age structured S.1. epidemic problem
in a heterogeneous environment ...................................... 307
G. Lumer
A general "isotropic" Paley-Wiener theorem and
some of its applications .............................................. 323
M. M astinsek
Asymptotic stability for a functional differential equation
in Hilbert space ..................................................... 333
U.F. Mayer and G. Simonett
Self-intersections for Willmore flow .................................. 341
O. G. Smolyanov, H. v. Weizsiicker and O. Wittich
Chernoff's theorem and the construction of semigroups .............. 349
K.-T. Sturm
Harmonic map heat flow generated by Markovian semigroups 359
J.A. van Casteren
The Hamilton-Jacobi-Bellman equation and
the stochastic Noether theorem ...................................... 375
A. Venni
Marcinkiewicz and Mihlin multiplier theorems,
and R-boundedness .................................................. 403
A. Visintin
Quasilinear P.D.E.s with memory operators.......................... 415
Preface
The seventh International Conference on Evolution Equations and their main areas
of Applications (where the emphasis evolves as time and problems change) was
held October 30 to November 4 at the CIRM (Centro Internazionale per la Ricerca
Matematica) in Trento, Italy.
In keeping with the basic principles and the recent tendencies governing these
International Conferences, it brought together many of the world's leading experts
in the fields mentioned, with particular effort on facilitating the interaction of
established scientists and emerging young promising researchers, as well as the
interaction of pure and applied specialists. In the latter directions, emphasis was
extended here to include in addition to Physical and Life Sciences, also Industry
and Economics.
Topics among the recent advances treated here concern new developments
in: moving boundary problems, asymptotics in non-linear Volterra equations and
other asymptotics related developments, Poincare inequality on stratified sets, time
operator and Markov processes in physics related advances, behavior of granu
lar matter, stochastic aspects of Hamilton-Jacobi-Bellman equation, very general
Paley-Wiener results applied to both classical and generalized functions, Ornstein
Uhlenbeck operators and processes, quasilinear PDEs with memory operators,
semi-group approach in economics (pricing theory) and other semi-group related
developments, convolution-evolution equation in aeroelasticity, new developments
in the study of age-structured models, new developments in maximal regularity.
This volume, while based on, and inspired by, the material presented at the
conference is, to be exact, not a mere "proceedings". The contributions submitted
in 2001 and 2002, the latest up to Fall 2002, contain also much material developed
after the conference by continuing research (incorporating additional new results),
also incorporating significant modifications suggested and required by the referees.
The volume also includes some up to date surveys of the state of the art in some
of the areas considered, refereed as well. Indeed the material has been rigorously
refereed (not all contributions submitted were accepted and in many cases more
or less ample modifications, requested by the referees ~ from inside or outside the
group of participants ~ were made).
The organizers thank the following institutions for providing support for the
conference:
• CIRM (Centro Internazionale per la Ricerca Matematica, Trento)
• EU (Grant HPCFCT-2000-00049: High Level Scientific Conferences Program)
• Mathematics Department, University of Trento
• INDAM-GNAFA (Istituto Nazionale di Alta Matematica, Gruppo Nazionale
Analisi Funzionale e Applicazioni).
viii Preface
We thank our colleagues and collaborators of the scientific committee, in par
ticular P. Dufour, M. Miranda, A. Valli, L. Weis for assistance and advice. We
thank L. Bouchez and A. Micheletti for technical and secretarial help. Finally, we
thank the contributors, referees, Birkhiiuser Verlag, particularly T. Hintermann
and T. Hempfling for their help and cooperation in making possible and producing
this volume, which we hope will in turn be valuable to the Scientific Community.
Trento, December 18, 2002 Mimmo Iannelli, Gunter Lumer
Progress in Nonlinear Differential Equations
and Their Applications, Vol. 55, 1-12
© 2003 Birkhauser Verlag Basel/Switzerland
Evolution Problems Associated to Linear
Growth Functionals: The Dirichlet Problem
Fuensanta Andreu, Vicent Caselles, and Jose M. Mazon
To the Memory of Ph. Benilan
1. Introduction and preliminaries
Let D be a bounded set in JRN with Lipschitz continuous boundary aD. We are
interested in the problem
aU
at = div a(x, Du) in Q = (0, (0) x D
{
u(t, x) = y(x) on S = (0,00) x aD (1.1)
u(O, x) = uo(x) in xED
f(~) = { ~ 1I~112
II~II- ~ if II~II;:::: 1
and the evolution problems associated with the Lagrangians:
f(x,~) = ,/1 + x2 + 11~112,
which was considered by S. Bernstein ([9]). On the other hand, problem (1.1) is
studied in [14J for some Lagrangians f, which do not include the nonparametric
area integrand, but instead include the plasticity functional and the total varia
tion flow, that is, the case f(~) = II~II. Now, the concept of solution given in [14J
is the one obtained by considering the abstract Cauchy problem in L2(D) associ
ated to the relaxed energy, but the subdifferential of the energy functional is not
characterized. For the particular case of the total variation flow, we give in [3J
a different approach to the Dirichlet problem. There, we studied the problem in
2 Fuensanta Andreu, Vicent Caselles, and Jose M. Mazon
the framework of the L1-theory, and we characterized the subdifferential in L2(0)
of its relaxed energy (we refer also to [2] where we treated the L1-theory for the
Neumann problem for the total variational flow).
In general, problem (1.1) does not have a classical solution. Our aim is to
introduce a concept of solution of the Dirichlet problem (1.1), for which existence
and uniqueness for initial data in Ll (0) is obtained. To do that, first we charac
terize the subdifferential of the energy associated with the problem and we use
the nonlinear semigroup theory to get existence and uniqueness of strong solution
when the initial data are in L2(0). After this, we use again the nonlinear semi
group theory to get existence of entropy solution and using the Kruzhkov's methos
of doubling variables both in space and time we get the uniqueness.
In order to consider the relaxed energy we recall the definition of function of
measure (see for instance, [7] or [12]). Let 9 : 0 X JRN ---t JR be a CaratModory
function such that
Ig(x,~)1 :::; M(l + II~II) (1.2)
for some constant M ;::: O. Furthermore, we assume that 9 possesses an asymptotic
function, i.e., for almost all x E 0 there exists the finite limit
lim tg(x,~) = gO(x,~). (1.3)
t .... O+ t
It is clear that the function gO(x,~) is positively homogeneous of degree one in ~,
i.e.,
for all x, ~ and s > O.
We denote by M(O, JRN) the set of all JRN -valued bounded Radon measures
on O. Given p E M(O, JRN), we consider its Lebesgue decomposition
p = pa + pS,
where pa is the absolutely continuous part of p with respect to de Lebesgue mea
sure I: N of JRN, and pS is singular with respect to I: N. We denote by pa (x) the
density of the measure pa with respect to I:N and by (dpS jdlpIS)(x) the density
of pS with respect to Ipls.
Given p E M(O, JRN), we define jl E M(O, JRN+l) by
jl(B):= (p(B),I:N(B)),
for every Borel set B C JRN. Then, we have
Hence, we have
