Semigroups of Linear and Nonlinear Operations and Applications: Proceedings of the Curaçao Conference, August 1992

279 Pages·1993·9.74 MB·English

by  Jerome A. Goldstein (auth.), Gisèle Ruiz Goldstein, Jerome A. Goldstein (eds.)

SEMIGROUPS OF LINEAR AND
NONLINEAR OPERATIONS AND APPLICATIONS
Semigroups of
Linear and No nlinear
Operations and Applications
Proceedings of the Cura~ao Conference,
August 1992
Edited by
GISELE RUIZ GOLDSTEIN
and
JEROME A. GOLDSTEIN
Department of Mathematics,
Louisiana State University
Baton Rouge, Louisiana, U.SA
SPRINGER SCIENCE+BUSINESS MEDIA, B.V.
A c.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-94-010-4834-7 ISBN 978-94-011-1888-0 (eBook)
DOI 10.1007/978-94-011-1888-0
Printed an acid-free paper
All Rights Reserved
© 1993 Springer Science+Business Media Dordrecht
Origina1ly published by Kluwer Academic Publishers in 1993
Softcover reprint of the hardcover 1 st edition 1993
No part of the material protected by this copyright notice may be reproduced ar
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information starage and
retrieval system, without written permission from the copyright owner.
Table of Contents
Director's Preface 3
Editors' Preface 7
Jerome A. Goldstein:
A Survey of Semigroups of Linear
Operators and Applications 9
Gisele Ruiz Goldstein:
Nonlinear Semigroups and Applications 59
Wim Caspers and Philippe Clement:
A Bifucation Problem for Point Interactions in Ll (IR?) 99
Alfonso Castro and Ratnasingham Shivaji:
Semipositone Problems 109
Iona Cioranescu:
A Generation Result for C-regularized Semigroups 121
Giuseppe Da Prato:
Smoothing Properties
of Heat Semi groups in Infinite Dimensions 130
W.E. Fitzgibbon, S.L. Hollis, and J.J. Morgan:
Locally Stable Dynamics for Reaction-Diffusion Systems 143
W. Fitzgibbon, M. Parrott, and Y. You:
Global Dynamics of Singularly
Perturbed Hodgkin-Huxley Equations 159
Matthias Hieber:
On Strongly Elliptic Differential Operators on Ll(JRn ) 177
Alessandra Lunardi:
Stability and Local Invariant Manifolds in
Fully Nonlinear Parabolic Equations 185
Adam C. McBride:
Fractional Integrals and Semigroups 205
Rainer Nagel:
Spectral and Asymptotic Properties of
Strongly Continuous Semigroups 225
1
2 Table of Contents
J. W. Neuberger:
Continuation for Quasiholomorphic Semigroups 241
Michel Pierre and Didier Schmitt:
Global Existence for a Reaction-Diffusion System
with a Balance Law 251
G. F. Webb:
Convexity of the Growth Bound of
Co-semigroups of Operators 259
Index 271
Director's Preface
This is the first publication which follows an agreement by
Kluwer Publishers with the Caribbean Mathematics Foundation
(CMF), to publish the proceedings of its mathematical activities.
To which one should add a disclaimer of sorts, namely that this
volume is not the first in a series, because it is not first, and be-
cause neither party to the agreement construes these publications
as elements of a series. Like the work of CMF, the arrangement
between it and Kluwer Publishers, evolved gradually, empirically.
CMF was created in 1988, and inaugurated with a conference
on Ordered Algebraic Structures. Every year since there have
been gatherings on a variety of mathematical topics: Locales and
Topological Groups in 1989; Positive Operators in 1990; Finite
Geometry and Abelian Groups in 1991; Semigroups of Operators
last year. It should be stressed, however that in preparing for
the first conference, there was no plan which might have augured
what came after. One could say that one thing led to another,
and one would be right enough.
Yet, that is not quite the whole story, For a long time it had
been clear to me that the Caribbean Basin and Latin America
languished in a general state of mathematical oblivion. And hav-
ing said that I'm reminded that one should be careful with such
statements, that there certainly are a number of distinguished
mathematicians from this part of the world, and also a number
of centers where first-rate work is being carried out. That is not
the point, however; what is the raison d'etre of CMF is that the
men and women and centers of distinction in mathematics in the
Caribbean Region and Latin America are few and far between;
that the vast majority of mathematicians there gradually lose
contact with the developed world because they are isolated in un-
derdevelopment, cut of from physical contact with experts. They
are isolated because their libraries are persistently out of date.
CMF enters this stage of underdevelopment with a mission
which is easy to formulate and yet difficult to carry out. The
mission is this: to do all it can to diminish the isolation alluded
3
G. R. Goldstein and J. A. Goldstein (eds.),
Semigroups ofL inear and Nonlinear Operations and Applications, 3-6.
© 1993 Kluwer Academic Publishers.
4 Director's Preface
to in the preceding paragraph. It is a difficult task for two main
reasons. First, it is not always clear how one can do the most
good. (More on this subject shortly.) And even when it is, there
are problems, faced by mathematicians of the region, which are
so typical of underdevelopment: poverty, institutional paternal-
ism and bureaucracy. We, in our cozy, developed universe, who
complain about the shortage of funds when it comes to travel
to conferences, should have some idea of the shortage faced by
mathematicians in the so-called Third World. As it seems in-
appropriate to use these pages to compile woes, let me illustrate
with one example: a mathematician from EI Salvador, Nicaragua,
Surinam or Guyana will typically get no funds at all from his/her
institution or home country, even when there is an explicit letter
of invitation.
But perhaps one need not strain to document poverty in un-
derdevelopment. Institutional paternalism, however, is less pal-
pable, and the effects of bureaucracy certainly as cruel as those
of poverty. The Spanish have a word for the kind of "red tape
chase" that takes days, weeks, months, and saps one's enthu-
siam: tramite. One Colombian colleague recently quipped that
his country was the land of "tramitologia".
In the developed world a mathematician who has made plans
to attend a conference simply arranges for a colleague to take over
his classes for a few days and leaves. Not so in the Caribbean
Region or Latin America, in general. There one has to submit
a formal request for leave, which then travels through the entire
bureaucratic apparatus of a university. If an academic goes to a
conference without official permission, he runs a great risk of not
being paid for the time away.
Such are the problems, even in the presence of the best laid
plans. The good news is that there are enough mathematicians in
the developed world with a true missionary spirit to challenge the
condition of underdevelopment in this region. It helps to be able
to invite one's colleagues to a tropical island like Cura~ao, where
the swimming and snorkeling are quite satisfactory, and where the
Dutch fa~ades on the St. Anna Bay strike the typical visitor with
more than a touch of wonder. And yet, not every distinguished
mathematician is suited to confront the special problems of un-
derdevelopment. Even the most altruistic soul has seen the reality
and been disappointed: at the low attendance by Caribbean and
Director's Preface 5
Latin American mathematicians at the workshops. I have seen
the disappointment in the faces of my colleagues, and shared in
it.
Which brings me round to the first-mentioned problem, in
discharging the simple mission of CMF. Deciding what themes
and topics to choose for the workshops and conferences.
Doubtless the attendance at these events by mathematicians
from the region is small because of the poverty and bureaucracy
to which I have already referred. However, one should also real-
ize that the region in question is huge, and that the interests of
its mathematicians varies enormously, not only in terms of disci-
pline or speciality, but also in terms of level of development. Last
year's workshop/conference on Semigroups of Operators and Ap-
plications is a good case in point. The conference attracted four
South American mathematicians: two from Chile and two from
Colombia. One might stare at that figure and say it is small.
My experience tells me that to draw four mathematicians from
South America constitutes an enormous success. The workshop
portion of the program drew another three participants from Suri-
nam and one Jamaican. What this number does not reveal is that
the latter four have become regulars at these events. Two of the
Surinamers have been attending since 1990, the third since 1991,
the Jamaican since the beginning.
One comes to an attitude of compromise and resignation,
while not losing sight of the goal, as stated above. Missionaries
of true spirit know how frustrating the task of promulgating the
Gospel can be. One has to know that the job is worth doing:
it is good to disseminate mathematics in the Caribbean Region
and Latin America. In the meantime one tries to find out how
best to serve its mathematical community, knowing that there
will be moments of frustration along the way; that one will make
mistakes along the way. One has to know that the crowds will
be small; that the participants who return, year after year, for
whatever reason, are, in the long run, the best promotors one has.
One counts one's blessings; the greatest two being to be able
to rely on the services of colleagues who instinctively understand
that the problem of doing something about underdevelopment is
not so simple, yet who by their talent and spirit contribute to the
realization of CMF's goals, with excellent programs. No one has
done this with more grace and verve than Jerry Goldstein and
6 Director's Preface
Gisele Ruiz Goldstein, through their program on Semigroups of
Operators and Applications. Generally, I have a solid expecta-
tion that the colleagues with whom I contract to prepare a work-
shop/conference for CMF will acquit themselves well. Indeed, I
can be more bold: I count on the people selected to present a
program under the auspices of CMF, which will produce an event
of the highest quality, and also display the kind of sensitivity one
needs when coming face to face with underdevelopment. I have
not yet been disappointed on either point.
During the three days of the workshop, the Goldsteins pre-
sented an account of both the linear and non-linear theories of
semi groups of operators, by clearly grounding the subject both
in terms of its origins in physics and of the philosophy of its de-
velopment, returning with regularity to a number of illustrative
examples. Perhaps it is not quite fair to volunteer myself as a
judge, but, as decidedly a non-expert in this field, I was able to
see the logic from formulation to execution, and understand, if
very few of the details, at least the mathematical tools that are
brought to bear. It seems to me that one cannot ask more of such
a workshop presentation.
As to the conference which followed, I can only judge by the
comments of its participants. I spoke to almost all of the twenty
or so, and each was thoroughly satisfied with the mathematical
event. We plan to reprise a conference on this discipline before
the end of the decade.
I spoke earlier of two blessings; I've expatiated on one, but
have not forgotten the other. One is fortunate to have in Dr.
David Larner and his staff at the Science and Technology Division
of Kluwer Academic Publishers people who have understood the
problem of disseminating mathematics in the region, and have
backed this understanding and their encouragement of CMF with
deeds. Their agreement to be the publishers of these proceedings
is but one of them.
Jorge Martinez, Director
Caribbean Mathematics Foundation
Gainesville, FL.
April 1993
Editors' Preface
The conference on Semi groups of Operators and Applications
took place in Curac;ao during August 1992. In the first week
the two editors gave short courses to the Caribbean Mathematics
Foundation, Gisele Goldstein's lectures being on nonlinear semi-
groups and Jerry Goldstein's being on linear semigroups. Those
short courses sketched the theory and emphasized the applica-
tions, especially to PDE. The second week was devoted to an
international conference, and lectures were given by some of the
leading people in the field.
Curac;ao is a paradise, and everyone, especially the under-
signed, was astonished by the breathtaking beauty of the is-
land. Jorge Martinez, the Director of the Caribbean Mathemati-
cal Foundation, is a masterful organizer. From his choice of hotel
and conference location through the elegant coffee breaks to the
conference banquet and other entertainment, he was the perfect
host. In particular, he arranged for perfect weather throughout
the meeting. We were all touched by Jorge's love for mathematics
and for the Caribbean region. All the participants had a wonder-
ful time.
The first two papers are written versions of the short courses
given by the editors. The audience was lively and attentive and
we are grateful for that. The rest of the papers are from the
international conference portion of the program.
Some of the lectures given at the conference appear here as
jointly authored papers. The corresponding lectures were pre-
sented by Alfonso Castro, Philippe Clement, Bill Fitzgibbon,
Mary Parrott, and Michel Pierre.
Also, several interesting lectures were given which do not
appear in these written proceedings. Those were:
J. Robert Dorroh: "Existence and regularity of solutions of
singular quasi-linear diffusion equations",
Klaus Engel: "On dissipative wave equations in reflexive Ba-
nach spaces" ,
7
G. R. Goldstein and 1. A. Goldstein (eds.),
Semigroups of Linear and Nonlinear Operations and Applications, 7-8.
© 1993 Kluwer Academic Publishers.