Miniconference on Operator Theory and Partial Differential Equations

Preface

This volume contains the proceedings of a three day mini-conference on operator theory and partial differential equations held at Macquarie University in September 1986, under the sponsorship of the Centre for Mathematical Analysis (Australian National University) whose financial assistance is gratefully acknowledged.  It is divided into two parts.  The first consists of reports of one hour invited talks.  The second is devoted to half hour research reports presented at the conference.

We gratefully acknowledge the support of Professor Neil Trudinger, director of the CM, and of the contributors to this volume, as well as the superb organizational and typing assistance of Marilyn Gray, Dorothy Nash, Joyce Heinze and Jill Smith.

Dedication

During 1986 Professor I. Kluvánek resigned his chair of mathematics at the Flinders University of South Australia.  His influence on the course of mathematics, at the teaching level and the research level, has been significant and inspiring both in Australia and overseas.  He has not only contributed fundamental works of his own but has supervised several graduate students who have gone on to successful mathematical careers.  Seven of these former students (B. Jefferies, G Knowles, C Meaney, S Morris, R Nillsen, S Okada and W Ricker) have made contributions to the proceedings of this mini-conference.  Other form students include A Geue, S McKee, M Sears and AK Whitford.

Igor Kluvánek obtained his first degree, in electrical engineering specializing in vacuum technology, from the Slovak Polytechnic University, Bratislava, in 1953.  His first appointment was in the Department of Mathematics of the same institution.  At the same time he worked for his C.Sc. degree obtained from the Slovak Academy of Sciences.  In the early 60s he joined the Department of Mathematical Analysis of the Šafárik University of Košice.  During 1967-68 he held a visiting position at Flinders University of South Australia.  The events of 1968 in Czechoslovakia and some unusual coincidences created circumstances making it impossible for him and his family to return to their homeland.  The Flinders University was able to create a chair in applied mathematics to which he was appointed in January 1969 and has occupied until his recent resignation.

Professor Kluvánek has made significant contributions to applied mathematics, functional analysis, operator theory and vector-valued integration.  One needs only to consult his book “Vector measures and control systems” written with Greg Knowles or examine the contents and historical notes of the monograph “Vector Measures” by J Diestel and JJ Uhl Jnr, to see that his penetrating studies into this area, of which he is one of the pioneers, pervade the subject.   He has also made important contributions to various topics on harmonic analysis.  For a sample of his influence in this area we refer to the excellent survey article “Five short stories about the cardinal series”, Bull American Mathematical Society 12(1985), 45-89 By JR Higgins which highlights the essential role played by the paper in the ‘story’ of the sampling theorem.  In Kluvánek introduced the concept of a closed vector measure.  This notion was crucial for his investigations of the range of a vector measure and led to the extension to infinite dimensional spaces of the classical Liapunov convexity theorem, together with many consequences and applications.  This work was in collaboration with G Knowles and settled many of the major problems in this area.  The notion of a closed vector measure stimulated much research, especially by W Graves and his students at Chapel Hill, North Carolina.  In recent years it has turned out that this notion is not only a basic tool in the study of algebras of operators generated by Boolean algebras of projections but lies at the very core of the major theorems in this area, even throwing a new perspective on a classical results in this field.

As successful as the theory of integration with respect to countably additive vector measures has been in various branches of mathematics, such as mathematical physics, functional analysis and operator theory, for example it is also known that there are fundamental problems which cannot be treated in this way.  Nevertheless, these problems still seem to require for their solution “some sort of integration process”.  As seen by the references [42.43,44] and the paper presented at this conference [45, it is clear that these problems are being vigorously pursued by Professor Kluvánek.  It seems that his resignation has not curtailed his enthusiasm for research.

As well as his research publications, it should be mentioned that Professor Kluvánek has co-authored with L Mišík and M Švec, a two volume text book (In Slovak) on Differential and Integral calculus, Analytic geometry, Differential equations and Complex variables which has seen two editions and been widely used in Czechosolvakia.  He also wrote lecture notes (in Slovak) with M. Kovárík on first year university analysis and a popular book (also in Slovak) with L Bukovskÿ on the pigeon-hole principle.  In addition, he has written various articles of a pedagogic nature.

Contents

Chapter Page no
1 - On continuation of quasi-analytic solutions of PDE's to compact convex sets (PDF, 718KB)
RF Bonner
1
2 - On Isomorphisms of Algebras of operators (PDF, 835KB)
Michael Cowling
11
3 - Sinular integrals and maximal functions on certain lie groups (PDF, 379KB)
GI Gawdry
21
4 - Integration for the spectral theory (PDF, 575KB)
Igor Kluvanek
26
5 - Estimates for linear systems of operator equations (PDF, 790KB)
Alan J Pryde
35
6 - A new approach to the schauder estimates for linear elliptic equations (PDF, 433KB)
Neil S Trudinger
52
7 - Solvability of differential operators I: Direct and semidirect products of lie groups (PDF, 1536KB)
FD Battesti
60
8 - Solvability of differential operators II: Semisimple lie groups (PDF, 822KB)
FD Battesti, AH Dooley
85
9 - Regularity theorems for elliptic equations with non-smooth coefficients (PDF, 724KB)
Huy-Qui Bui
95
10 - Stable harmonic maps with values in PMC (PDF, 365KB)
D Burns and P de Bartolomeis
111
11 - Trinity....A tale of three cardinals (PDF, 579KB)
Joan Cleary, Sidney A Morris
117
12 - Holomorphic representations of SL (2.R) and Qantum scattering theory (PDF, 746KB)
JV Corbett
128
13 - Fourier transform of surface carried measures (PDF, 378KB)
MG Cowling, G Mauceri
142
14 - The first boundary value problem for solutions of degenerate quasilinear parabolic equations (PDF, 578KB)
Guang-Chang Dong
148
15 - Eigenstructure specificaition via state boundary feedback for linear systems in hilbert space (PDF, 520KB)
WF Holland
158
16 - On a modified mean curvature flow (PDF, 509KB)
Gerhard Huisken
169
17 - Eigen-expasions of some schrodinger operators and nilpotent lie groups (PDF, 547KB)
Andrzej Hulanicki, Joe W Jenkins
176
18 - Bilinear integrals and radon-Nikodym derivaties (PDF, 1229KB)
BRF Jefferies
185
19 - Asymptotic limits in multi-phase systems (PDF, 424KB)
Greg Knowles
201
20 - Operators which have a H(infinity) functional calculus (PDF, 1229KB)
Alan McIntosh
210
21 - Invariant differential operators and representation theory (PDF, 1126KB)
Christopher Meaney
232
22 - Some basic sequences and their moment operators (PDF, 2048KB)
Rodney Nillsen
246
23 - On the existance of positive entire solutions of non-linear elliptic problems in IR(n) (PDF, 485KB)
Ezzat S Noussair
272
24 - Integration of conuclear space valued functions (PDF, 588KB)
Susumu Okada
281
25 - Regularity of generalized scalar operators with spectrum contained in a line (PDF, 333KB)
Werner Ricker
291
26 - Commutator theory and partial differential operators on hilbert space (PDF, 399KB)
Derek W Robinson
295
27 - Some positive eigenfunctions for elliptic operators with oblique derivative boundary conditions and consequences for the stationary densitiies of relfected brownian motions (PDF, 72KB)
Ruth J Williams
326
28 - Remarks on 2nd order elliptic systems in lipschitz domains (PDF, 1331KB)
GC Verchota
303
29 - Evolution Equations of parabolic type (PDF, 413KB)
A Yagt
327
30 - Banach spaces with many projections (PDF, 1024KB)
David Yost, Brailey Sims
335

Printed copy

Price: $27.50 Buy now

Volume

Volume 14

Date

8 Sep 1986 to 10 Sep 1986

Venue

Macquarie University, Ryde NSW

Updated:  28 April 2017/Responsible Officer:  Director/Page Contact:  School Manager