The Boundary Regularity of Minimal Surfaces
In this chapter we deal with the boundary behaviour of minimal surfaces with particular emphasis on the behaviour of stationary surfaces at their free boundaries. This and the following chapter will be the most technical and least geometric parts of our l
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Grundlehren der mathematischen Wissenschaften 296 ASeries 0/ Comprehensive Studies in Mathematics
Editors M. Artin S. S. Chern 1. Coates 1. M. Fröhlich H. Hironaka F. Hirzebruch L. Hörmander S. MacLane C. C. Moore 1. K. Moser M. Nagata W. Schmidt D. S. Scott Ya. G. Sinai 1. Tits M. Waldschmidt S.Watanabe
Managing Editors M. Berger B. Eckmann S. R. S. Varadhan
Ulrich Dierkes Stefan Hildebrandt Albrecht Küster Ortwin Wohlrab
Minimal Surfaces 11 Boundary Regularity
With 59 Figures and 4 Colour Plates
Springer-Verlag Berlin Heidelberg GmbH
Ulrich Dierkes Stefan Hildebrandt Albrecht Küster Universität Bonn, Mathematisches Institut Wegeierstraße 10, D-5300 Bonn, Federal Republic ofGermany Ortwin Wohlrab Mauerseglerweg 3, D-5300 Bonn, Federal Republic ofGermany
Mathematics Subject Classification (1991): 53A 10, 35J60
ISBN 978-3-662-08778-7
Library ofCongress Cataloging-in-Publication Data Minimal surfaceslUlrich Dierkes... [etaI.] v. cm. - (Grundlehren der mathematischen Wissenschaften; 295-296) Includes bibliographical references and indexes. Contents: I. Boundary value problems - 2. Boundary regularity. ISBN 978-3-662-08778-7 ISBN 978-3-662-08776-3 (eBook) DOI 10.1007/978-3-662-08776-3 I. Surfaces, Minimal. 2. Boundary value problems. I. Dierkes, Ulrich. 11. Series. QA644.M56 1992 516.3'62 - dc20 90-27155 CIP This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concemed, specificallythose oftranslation, reprinting, reuse ofillustrations, recitation, broadcasting, reproduction on microfilms or in other way, and storage in data banks. Duplication ofthis publication or parts thereofis permitted only under the provisions ofthe German Copyright Law ofSeptember 9, 1965, in its current version, and a copyright fee must always be obtained from Springer-Verlag Berlin Heidelberg GmbH. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1992
OriginaJly published by Springer-Verlag Berlin Heidelberg New Yoti0
whence we infer that VYh tends uniformly to z on every Q c c BR • Together with the uniform convergence of Yh to Y on Q c c BR as h -+ 0 we infer that Y E Cl (BR ) and Vy(w) = z(w) for any w E BR • Consequently
ly(w)1 + IVwY(w) I ~
LR {IH(w, ()I + IVwH(w, ()I} Iq(()ld
~ c(b, R)a
2(
for all w E BR ;
thus (13) is also verified. Now let Wl' W2 E BR , and set p:= IWl - w21. Then we infer from (15) that
7.1 Potential-Theoretic Preparations
11
Note that
and the mean value theorem implies
oH I 2bp loH ou (w 1, 0 - oU (W2' 0 ~ Iw* _ ,,2
for some w* = (1 - t)w 1 + tw2, 0< t < 1. If I( - w1 1;::: 2p, we infer that I( - w*1 ;::: I( - wll-lw l - w*1 ;:::
tl( -
wll,
and therefore
oH oH I 8bp (wl, 0 - ou (w 0 ~ I( _ wI12 lau 2,
for
i' - wd;::: 2p.
Thus we arrive at
:s;: ab[4np
+ 6np + 16np log~J :s;: ac(b, R, J-L)pl'
for any J-L E (0, 1) and p = IW1 - w2 1, and (14) is proved. Estimates (13) and (14) imply that y can be extended to ER as a function of dass C I 'I'(ER) for any J-L E (0,1). 0
Lemma
