The Minkowski Multidimensional Problem

116 Pages·1978·5.197 MB·English

by  Aleksei V. Pogorelov

THE MINKOWSKI
MULTIDIMENSIONAL
PROBLEM
A. V. Pogorelov
with an introduction by
Louis Nirenberg
THE MINKOWSKI
MULTIDIMENSIONAL
PROBLEM
ALEKSEY V. POGORELOV
The Minkowski problem is that
of finding a closed convex surface
in R3 whose Gaussian curvature is a
given positive function K of the
exterior unit normal; K necessarily
satisfies the three conditions
/M T1 (ö*o(ö = 0
/=1,2,3
where integration is over the unit
sphere.
In addition to the regular solu­
tion of the Minkowski problem, the
book covers a number of related
questions in geometry and the
theory of differential equations
with partial derivatives. In particu­
lar, it considers the general problem
of a closed convex hypersurface
with a prescribed function of curva­
ture of any degree. The generalized
solutions of the multidimensional
analogue of the Monge-Ampere
equation are studied and their regu­
larity under certain conditions is
proved. The Dirichlet problem is
solved. Also, improper convex
affine hyperspheres are discussed
and proved to be elliptic para­
boloids if they are full.
In the words of the introducer,
Professor Louis Nirenberg of the
Courant Institute of Mathematical
Sciences, “This book will introduce
to the reader some beautiful geo­
metric problems and a variety of
interesting, deep, techniques for ob­
taining a priori estimates for elliptic
(continued on inside back flap)
Li
THE MINKOWSKI
MULTIDIMENSIONAL
PROBLEM
SCRIPTA SERIES IN MATHEMATICS
Tikhonov and Arsenin • Solutions of Ill-Posed Problems, 1977
Rozanov • Innovation Processes, 1977
Pdgorelov • The Minkowski Multidimensional Problem, 1978
Kolchin, Sevast'yanov, and Chist'yakov • Random Allocations, 1978
Boltyanskiy • Hilbert’s Third Problem, 1978
THE MINKOWSKI
MULTIDIMENSIONAL
PROBLEM
Aleksey Vasu/yevich Pogorelov
USSR Academy of Sciences
Translated by Vladimir Oliker, University of Iowa and
introduced by Louis Nirenberg, Courant Institute of
Mathematical Sciences
1978
V. H. WINSTON & SONS
Washington, D.C.
A HALSTED PRESS BOOK
JOHN WILEY & SONS
New York Toronto London Sydney
Copyright © 1978, by V. H. Winston & Sons, a Division of
Scripta Technica, Inc.
All rights reserved. Printed in the United States of America.
No part of this publication may be reproduced, stored in a
retrieval system, or transmitted, in any form or by any means,
electronic, mechanical, photocopying, recording, or otherwise,
without prior written permission of the publisher.
V. H. Winston & Sons, a Division of Scripta Technica, Inc.,
Publishers
1511 K Street, N.W., Washington, D.C. 20005
Distributed solely by Halsted Press, a Division of John Wiley
& Sons, Inc.
Library of Congress Cataloging in Publication Data
Pogorelov, Aleksei Vasil’evich.
The Minkowski multidimensional problem.
(Scripta series in mathematics)
Translation of Mnogomemaía problema Minkovskogo.
Bibliography: p.
I. Convex surfaces. 2. Spaces, Generalized.
I. Title. II. Series.
QA643.P73813 516\362 77-16680
ISBN 0-470-99358-8
Composition by Isabelle Sneeringer, Scripta Technica, Inc.
CONTENTS
INTRODUCTORY COMMENTARY....................................... I
PREFACE TO THE AMERICAN EDITION ........................ 3
FOREWORD................................................................................ 5
INTRODUCTION ...................................................................... 7
§1. Convex Bodies and Hypersurfaces in En .................. 9
§2. Generalized Solution of the Minkowski Problem . . 22
§3. Regular Solution of Minkowski’s Problem ............. 32
§4. Generalization of Minkowski’s Problem .................. 50
§5. Multidimensional Analog of the Monge-Ampère
Equation ................................................................... 66
§6. On Improper Convex Affine Hyperspheres ........... 89
BIBLIOGRAPHY ................................................................... 105
V

INTRODUCTORY COMMENTARY
The Minkowski problem is that of finding a closed convex
surface in R3 whose Gaussian curvature is a given positive function
K of the exterior unit normal; K necessarily satisfies the three
conditions
J&K“1 «)Л0(8 = 0 /=1,2,3
where integration is over the unit sphere.
In case A' is a smooth function the problem has a smooth
solution and a presentation of this result is contained in the, now
classic, book [8] by the author. The Minkowski problem has a
direct extension for closed convex hypersurfaces in Rn. However,
after the result in R39 it still took a number of years before the
corresponding higher dimensional result was established—in a series
of papers by A.V. Pogorelov (in the bibliography). This book
presents a detailed treatment of the results described in those
papers as well as generalizations of the Minkowski problem.
Analytically, the Minkowski problem involves solving a highly
nonlinear partial differential equation of Monge-Ampère type. An
example of such an equation, for a convex function u(xl,... ,xn)
I
2 INTRODUCTORY COMMENTARY
defined in a domain in Rn is
where ф(х) is a given position function. To prove the existence of
solutions of such equations, satisfying some boundary conditions,
is no easy task. It involves finding a priori estimates for solutions
and their derivatives up to third order. These estimates are all due
to the author. In deriving bounds for the third derivatives he
makes use of ideas of E. Calabi in [10]. This book also contains
Pogorelov’s proof of the conjecture (established previously for n <
5), that the only convex function satisfying (*) in all of Rn with ф
= I is a quadratic.
This book will introduce to the reader some beautiful geometric
problems and a variety of interesting, deep, techniques for obtain­
ing cl priori estimates for elliptic equations—written by a master in
the subject. It should serve as a basic reference for many years.
Attention of the reader should also be drawn to the recent
papers by S. Y. Cheng and S. T. Yau on these topics:
On the regularity of the solution of the и-dimensional Minkowski
problem, Comm. Pure Appl Math., (1976), 29, pp. 495-516.
On the regularity of the Monge-Ampère equation
det = F(x, u)
\Ъх1ЪхЧ v f
Comm. Pure AppL Math, (1977), 30, pp. 41-68.
Louis Nirenberg
Courant Institute of Mathematical Sciences
November 1977