Selected Paper s o n Differential Equation s and Analysi s

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American Mathematica l Societ y

TRANSLATIONS Series 2 • Volum e 21 5

Selected Paper s o n Differential Equation s and Analysi s

•s\\& America n Mathematica l Societ y Providence, Rhod e Islan d

http://dx.doi.org/10.1090/trans2/215

2000 Mathematics Subject Classification. Primar y 46L37 , 81T05 , 60J65 , 31A05 , 31B05 , 31B25, 31C35 , 58Jxx , 62D05 , 35K57 , 35R35 , 42B25.

Library o f Congres s Cataloging-in-Publicatio n Dat a

Selected paper s o n differentia l equation s an d analysis . p. cm . (America n Mathematica l Societ y translations , ISS N 0065-929 0 ; ser. 2 , v . 215 )

"[Translations o f articles ] originall y publishe d i n th e journa l Sugaku"—Foreword . Includes bibliographica l references . ISBN 0-8218-3927- 6 (acid-fre e paper ) 1. Differentia l equations . 2 . Functiona l analysis . 3 . Operato r algebras . 4 . Potentia l theor y

(Mathematics). I . Sugaku . II . Series .

QA3.A572 ser . 2 , vol . 21 5 [QA370] 510 s—dc2 2 [515'.35] 2005052407

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Contents

Foreword

Regularity i n fre e boundar y problem s GEORG S . WEIS S

Behavior o f spatial critica l point s an d zero s of solutions o f diffusion equation s SHIGERU SAKAGUCH I

Martin boundar y an d boundar y Harnac k principl e fo r non-smoot h domain s HlROAKI AlKAW A

Singular integral s an d Littlewood-Pale y function s SHUICHI SAT O

Convergence o f metri c measur e space s an d energ y form s ATSUSHI KASU E

Subfactor theor y an d it s applications : Operato r algebra s an d quantu m field theory YASUYUKI KAWAHIGASH I

Brownian motio n an d valu e distributio n theor y o f holomorphi c map s an d harmonic map s ATSUSHI ATSUJ I

Statistical inferenc e i n two-stag e samplin g MAKOTO AOSHIM A

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Foreword

This i s a collection o f several papers originally publishe d i n the journal Sugaku. Normally thes e translate d article s woul d hav e appeare d i n th e America n Mathe -matical Societ y journa l Sugaku Expositions. T o spee d u p publication , th e AM S decided t o publis h the m a s a volum e i n it s serie s America n Mathematica l Societ y Translations Serie s 2 .

The volum e begin s with th e articl e "Regularit y i n free boundar y problems " b y Georg S . Weiss , devote d t o partia l differentia l equation s wit h fre e boundary . Th e author describe s thre e particula r fre e boundar y problems : th e obstacl e problem , the hea t equatio n wit h stron g absorption , an d a singula r limi t proble m i n combus -tion theory . Fo r eac h o f thes e problem s th e autho r present s recen t result s relate d to regularit y condition s o f wea k solution s o f th e correspondin g equation s and , i n particular, t o conditions ensurin g th e smoothnes s o f the boundar y betwee n region s corresponding t o differen t phases .

The article "Behavio r of spatial critica l points and zero s of solutions of diffusio n equations" b y Shigeru Sakaguch i i s devoted to properties of solutions of Cauchy an d Neumann-type problem s fo r th e hea t equation s wit h N spatia l variables . Defin e the se t o f ho t spot s M(t) o f a solutio n u(y, i) o f such a n equatio n b y

M(t) = {x e R N : u(x, t) = ma x u(y, t)} . y<ERN

The articl e present s result s abou t th e evolutio n o f th e se t M(t) a s t — > oo. I n particular, th e autho r describe s th e case s whe n th e origi n i s a stationar y ho t spo t of a solutio n o f the Dirichle t proble m i n a bounde d domai n i n 1SL N.

The articl e "Marti n boundar y an d boundar y Harnac k principl e fo r non-smoot h domains" b y Hiroaki Aikawa presents results generalizing the classica l Poisson inte -gral representation o f a harmonic functio n o f the uni t bal l in the n-dimensiona l bal l to domains with non-smooth boundaries . Fo r such domains, the ordinary boundar y of a smooth domai n i s replaced b y the so-calle d minima l Marti n boundary , an d th e classical Herglotz' s theore m give s th e Martin integral representation, whic h i s a n analog o f th e Poisso n integral . Th e autho r studie s a clas s o f non-smoot h domain s for which the Martin boundar y ca n be described explicitl y and clarifie s th e relation -ship betwee n th e geometri c structur e o f a domai n an d th e propertie s o f harmoni c functions i n it .

In the articl e "Singula r integral s and Littlewood-Pale y functions" , Shuich i Sat o studies oscillatin g singula r integra l operator s wit h kernel s satisfying certai n growt h and smoothnes s condition s an d survey s som e result s o n th e boundednes s o f suc h operators i n various L p spaces .

vii

viii F O R E W O R D

In th e articl e "Convergenc e o f metri c measur e space s an d energ y forms" , Atsushi Kasu e consider s th e Dirichle t proble m fo r harmoni c function s an d stud -ies convergenc e o f the direc t calculu s o f variations method s fo r obtainin g solution s using th e genera l convergenc e propertie s o f Riemannia n manifold s withi n th e gen -eral theor y o f Hausdorff-Gromo v distanc e betwee n metri c spaces . Th e autho r als o studies th e analogou s convergenc e o f Riemannian vecto r bundles .

The article "Subfacto r theor y and its applications: Operato r algebra s and quan -tum field theory" b y Yasuyuki Kawahigash i provide s a review of recent application s of the theor y o f operator algebra s i n geometry an d physics . Th e autho r start s wit h the presentatio n o f result s b y Alai n Conne s o n classificatio n o f factor s an d o n th e use of operator algebra s as one of the main ingredients of Connes's approac h t o non-commutative geometry . The n h e describe s th e application s o f operato r algebra s i n the classificatio n o f knots (subfacto r theor y o f V. F. R. Jones ) an d i n quantum field theory.

In th e articl e "Brownia n motio n an d valu e distributio n theor y o f holomorphi c maps an d harmoni c maps" , Atsush i Atsuj i give s a surve y o f result s relatin g th e Brownian motion on a Riemannian manifol d M t o the function theor y on M. Thes e properties ca n be viewed a s a generalization o f the so-called conforma l invarianc e of the Brownia n motion . Th e articl e start s wit h a presentatio n o f th e Davi s theore m about a typica l homotopi c clas s o f a Brownian pat h o n the plane . The n th e autho r describes recen t result s o n th e relationshi p betwee n Picard-typ e theorem s o n M and martingales , a s well as relations betwee n th e Nevanlinn a theor y an d stochasti c calculus.

Finally, th e articl e "Statistica l inferenc e i n two-stag e sampling " b y Makot o Aoshima discusses the two-stage sampling method i n hypothesis testing. Th e articl e starts b y describin g th e genera l two-stag e schem e o f hypothesi s testin g an d it s asymptotic an d statistica l properties . The n th e autho r continue s b y presentin g examples o f how suc h a scheme i s used i n practica l problem s o f hypothesis testing .

American Mathematica l Societ y

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Titles i n Thi s Serie s

215 Selecte d Paper s o n Differentia l Equation s an d Analysi s

214 N . N . Uraltseva , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society ,

Volume X

213 Ernes t Vinberg , Editor , Li e Group s an d Invarian t Theor y

212 V . M . Buchstabe r an d I . M . Krichever , Editors , Geometry , Topology , an d Mathematical Physic s

211 K . Nomizu , Editor , Selecte d Paper s o n Analysi s an d Differentia l Equation s

210 S . G . Gindikin , Editor , Li e Group s an d Symmetri c Spaces . I n Memor y o f F . I . Karpelevich

209 N . N . Uraltseva , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society ,

Volume I X

208 M . V . Karasev , Editor , Asymptoti c Method s fo r Wav e an d Quantu m Problem s

207 Yu . M . Suhov , Editor , Analyti c Method s i n Applie d Probabilit y

206 M . A . Shubi n an d M . S . Agranovich , Editors , Partia l Differentia l Equation s (Mar k

Viskik's Seminar )

205 N . N . Uraltseva , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society ,

Volume VII I

204 K . Nomizu , Editor , Selecte d Paper s o n Classica l Analysi s

203 N . N . Uraltseva , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society ,

Volume VI I

202 V . Turae v an d A . Vershik , Editors , Topology , Ergodi c Theory , Rea l Algebrai c

Geometry. Rokhlin' s Memoria l

201 Michae l Semenov-Tian-Shansky , Editor , L . D . Faddeev' s Semina r o n Mathematica l

Physics

200 L . Lerman , G . Polotovskii , an d L . Shilnikov , Editors , Method s o f Qualitativ e

Theory o f Differentia l Equation s an d Relate d Topic s

199 N . N . Uraltseva , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society ,

Volume V I

198 R . A . Minlos , Seny a Shlosman , an d Yu . M . Suhov , Editors , O n Dobrushin' s Way .

From Probabilit y Theor y t o Statistica l Physic s

197 Vladimi r Arnold , Max i m Kontsevich , an d Anto n Zorich , Editors , Pseudoperiodi c

Topology

196 Ya . Eliashberg , D . Fuchs , T . Rat iu , an d A . Weinstein , Editors , Norther n

California Symplecti c Geometr y Semina r

195 O . V . Belegrade k e t al. , Mode l Theor y an d Application s 194 A . Astashkevic h an d S . Tabachnikov , Editors , Differentia l Topolog y

Infinite-Dimensional Li e Algebras , an d Application s (D . B . Fuchs ' 60t h Anniversar y

Collection)

193 N . N . Uraltseva , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society ,

Volume V

192 Le v Beklemishev , Mat i Pentus , an d Nikola i Vereshchagin , Provability ,

Complexity, Grammar s

191 A . Yu . Morozo v an d M . A . Olshanetsky , Editors , Mosco w Semina r i n

Mathematical Physic s

190 S . Tabachnikov , Editor , Differentia l an d Symplecti c Topolog y o f Knot s an d Curve s

189 V . Buslaev , M . Solomyak , an d D . Yafaev , Editors , Differentia l Operator s an d Spectral Theor y (M . Sh . Birman' s 70t h Anniversar y Collection ) 188 O . A . Ladyzhenskaya , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society . Volume I V

TITLES I N THI S SERIE S

187 M . V . Karasev , Editor , Coheren t Transform , Quantization , an d Poisso n Geometr y

186 A . Khovanskii , A . Varchenko , an d V . Vassiliev , Editors , Geometr y o f Differentia l

Equations

185 B . Feigi n an d V . Vassiliev , Editors , Topic s i n Quantu m Group s an d Finite-Typ e

Invariants (Mathematic s a t th e Independen t Universit y o f Moscow )

184 Pete r Kuchmen t an d Vladimi r Lin , Editors , Voronez h Winte r Mathematica l School s

(Dedicated t o Seli m Krein )

183 K . Nomizu , Editor , Selecte d Paper s o n Harmoni c Analysis , Groups , an d Invariant s

182 V . E . Zakharov , Editor , Nonlinea r Wave s an d Wea k Turbulenc e

181 G . I . Olshanski , Editor , Kirillov' s Semina r o n Representatio n Theor y

180 A . Khovanskii , A . Varchenko , an d V . Vassiliev , Editors , Topic s i n Singularit y

Theory

179 V . M . Buchstabe r an d S . P . Novikov , Editors , Solitons , Geometry , an d Topology :

On th e Crossroa d

178 V . Kreinovic h an d G . Mints , Editors , Problem s o f Reducin g th e Exhaustiv e Searc h

177 R . L . Dobrushin , R . A . Minlos , M . A . Shubin , an d A . M . Vershik , Editors ,

Topics i n Statistica l an d Theoretica l Physic s (F . A . Berezi n Memoria l Volume )

176 E . V . Shikin , Editor , Som e Question s o f Differentia l Geometr y i n th e Larg e

175 R . L . Dobrushin , R . A . Minlos , M . A . Shubin , an d A . M . Vershik , Editors ,

Contemporary Mathematica l Physic s (F . A . Berezi n Memoria l Volume )

174 A . A . Bolibruch , A . S . Merkur'ev , an d N . Yu . Net sve taev , Editors , Mathematic s

in St . Petersbur g

173 V . Kharlamov , A . Korchagin , G . Polotovskii , an d O . Viro , Editors , Topolog y o f

Real Algebrai c Varietie s an d Relate d Topic s

172 K . Nomizu , Editor , Selecte d Paper s o n Numbe r Theor y an d Algebrai c Geometr y

171 L . A . Bunimovich , B . M . Gurevich , an d Ya . B . Pesin , Editors , Sinai' s Mosco w

Seminar o n Dynamica l System s

170 S . P . Novikov , Editor , Topic s i n Topolog y an d Mathematica l Physic s

169 S . G . Gindiki n an d E . B . Vinberg , Editors , Li e Group s an d Li e Algebras : E . B .

Dynkin's Semina r

168 V . V . Kozlov , Editor , Dynamica l System s i n Classica l Mechanic s

167 V . V . Lychagin , Editor , Th e Interpla y betwee n Differentia l Geometr y an d Differentia l

Equations

166 O . A . Ladyzhenskaya , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society ,

Volume II I

165 Yu . Ilyashenk o an d S . Yakovenko , Editors , Concernin g th e Hilber t 16t h Proble m

164 N . N . Uraltseva , Editor , Nonlinea r Evolutio n Equation s

163 L . A . Bokut' , M . Hazewinkel , an d Yu . G . Reshetnyak , Editors , Thir d Siberia n

School "Algebr a an d Analysis "

162 S . G . Gindikin , Editor , Applie d Problem s o f Rado n Transfor m

161 K . Nomizu , Editor , Selecte d Paper s o n Analysis , Probability , an d Statistic s

160 K . N o m i z u , Editor , Selecte d Paper s o n Numbe r Theory , Algebrai c Geometry , an d

Differential Geometr y

159 O . A . Ladyzhenskaya , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society , Volume I I

For a complet e lis t o f t i t le s i n thi s series , visi t th e AMS Bookstor e a t w w w . a m s . o r g / b o o k s t o r e / .