Analysis, partial differential equations and applications: The V.Maz'ya anniversary

342 Pages·2009·1.95 MB·English

by  Alberto Cialdea

Operator Theory: Advances and Applications
Vol. 193
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Analysis,
Partial Differential
Equations
and Applications
The Vladimir Maz’ya Anniversary Volume
Alberto Cialdea
Flavia Lanzara
Paolo Emilio Ricci
Editors
Birkhäuser
Basel · Boston · Berlin
Authors:
Alberto Cialdea Flavia Lanzara
Dipartimento di Matematica e Dipartimento di Matematico “Guido Castelnuovo”
Informatica Sapienza Università di Roma
Università della Basilicata Piazzale Aldo Moro 2
Viale dell’Ateneo Lucano 10 00185 Rome, Italy
85100 Potenza, Italy e-mail: [email protected]
e-mail: [email protected]
Paolo Emilio Ricci
Dipartimento di Matematico “Guido Castelnuovo”
Sapienza Università di Roma
Piazzale Aldo Moro 2
00185 Rome, Italy
e-mail: [email protected]
2000 Mathematical Subject Classification: 31-06, 35-06, 46-06
Library of Congress Control Number: 2009931267
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Contents
A. Cialdea, F. Lanzara and P.E. Ricci
On the Occasion of the 70th Birthday of Vladimir Maz’ya ........... ix
Yu. Burago and N.N. Kosovsky
Boundary Trace for BV Functions in Regions
with Irregular Boundary ............................................. 1
J. Burnett, O. Chervova and D. Vassiliev
Dirac Equation as a Special Case of Cosserat Elasticity .............. 15
I. Capuzzo Dolcetta
H¨older and Lipschitz Estimates for Viscosity Solutions
of Some Degenerate Elliptic PDE’s .................................. 31
A. Cialdea
Criteria for the Lp-dissipativity of PartialDifferential Operators ..... 41
A. Cianchi
Sharp Estimates for Nonlinear Potentials and Applications ........... 57
M. Frazier and I. Verbitsky
Solvability Conditions for a Discrete Model
of Schro¨dinger’s Equation ............................................ 65
Yu.I. Karlovich
An Algebra of Shift-invariant Singular Integral Operators
with Slowly Oscillating Data and Its Application to Operators
with a Carleman Shift ............................................... 81
C.O. Kiselman
Frozen History: Reconstructing the Climate of the Past .............. 97
G. Kresin
Multidimensional Harmonic Functions Analogues of Sharp
Real-part Theorems in Complex Function Theory .................... 115
vi Contents
F. Lanzara and G. Schmidt
Cubature of Integral Operators by Approximate
Quasi-interpolation .................................................. 129
S. Mayboroda and V. Maz’ya
Pointwise Estimates for the Polyharmonic Green Function
in General Domains ................................................. 143
R. McOwen
On Elliptic Operators in Nondivergence and in Double
Divergence Form .................................................... 159
D. Mitrea and M. Mitrea
On the Well-posedness of the Dirichlet Problem in
Certain Classes of Nontangentially Accessible Domains ............... 171
S. Molchanov and B. Vainberg
On Negative Spectrum of Schro¨dinger Type Operators ............... 197
G. Moscariello, A. Passarelli di Napoli and C. Sbordone
ACL-homeomorphisms in the Plane .................................. 215
D. Natroshvili and Z. Tediashvili
Crack Problems for Composite Structures ............................ 227
Y. Pinchover and K. Tintarev
On Positive Solutions of p-Laplacian-type Equations ................. 245
J. Rossmann
Mixed Boundary Value Problems for Stokes and
Navier-Stokes Systems in Polyhedral Domains ....................... 269
S. Samko
On Some Classical Operators of Variable Order
in Variable Exponent Spaces ......................................... 281
M.A. Vivaldi
Irregular Conductive Layers ......................................... 303
W.L. Wendland
On the Double Layer Potential ...................................... 319
Vladimir Maz’ya
OperatorTheory:
Advances andApplications,Vol.193, ix–xvii
(cid:2)c 2009Birkh¨auserVerlagBasel/Switzerland
On the Occasion of the 70th Birthday
of Vladimir Maz’ya
Alberto Cialdea, Flavia Lanzara and Paolo E. Ricci
This volume includes a selection of lectures given at the International Workshop
“Analysis, Partial Differential Equations and Applications”, held at the Mathe-
maticalDepartmentofSapienzaUniversity(Rome,June 30th–July3rd,2008),on
the occasion of the 70th birthday of Vladimir Maz’ya.
BesidesItaly,twentysevencountrieswererepresentedthere:Belarus,Canada,
China, Colombia, Croatia, Czech Republic, Finland, France, Georgia, Germany,
Greece, Israel, Mexico, New Zealand, Poland, Portugal, Rumania, Russia, Saudi
Arabia, South Korea, Spain, Sweden, Taiwan, The Netherlands, Turkey, United
Kingdom, and United States of America.
It is not surprising that the decision of the Italian National Institute for
AdvancedMathematics“F.Severi”(INDAM)todedicateaWorkshoptoVladimir
Maz’yawascrownedbysuchgreatsuccess.Thescientific andhumanendowments
of Maz’ya are well known.
HehasinspirednumerousresearchersinAnalysisanditsapplications,among
themmanyinItaly.Maz’yagladlyacknowledgesthatthisinspirationhasbeenmu-
tual. The ItalianschoolofAnalysis and PDEshas playedanimportantrole in his
development, starting with his undergraduate years 1955–1960and continuing to
this day. As a third year student, through S. Mikhlin’s lectures, he became ac-
quainted with Tricomi’s pioneering work on multi-dimensional singular integrals
[37], [38], a topic of Maz’ya’s keen interest in the future ([14], [21] and others). A
year later, Vladimir discovered the equivalence of various Sobolev type inequal-
ities with isoperimetric and isocapacitary inequalities, which strongly influenced
functional analysis and partial differential equations in subsequent years. In par-
ticular, he found the sharp constant in the E. Gagliardo inequality between the
Ln/(n−1) normofafunctionandtheL1normofitsgradient[10].LaterGagliardo’s
resultsonboundarytracesofSobolevfunctionsweredevelopedbyMaz’yaandhis
colleagues in various directions (see, for example, [24], [33], [34]).
FollowingMikhlin’srecommendation,Maz’yareadtheRussiantranslationof
CarloMiranda’s“EquazionialleDerivateParzialidiTipoEllittico”[36],whichhad
appearedin1957inMoscow.ThiscomprehensivesurveyoftheItaliancontribution
x A. Cialdea, F. Lanzara and P.E. Ricci
to the field, which at that time was undergoing a major expansion, became the
first book on PDEs to be read by the young Maz’ya. Miranda’s book strongly
influenced the shaping of Vladimir’s professional interests. An evidence to this is
his first publication which appeared in [9] exactly 50 years ago.
The year 1957 saw the appearance of the seminal article by E. De Giorgi on
the H¨older regularity of solutions to elliptic second-order equations with measur-
ableboundedcoefficients,whichhadatremendousimpactonthetheoryofPDEs,
notleasttheworkofMaz’ya.Inthearticle[11]of1961,hesolvedaproblemposed
by G. Stampacchia on an estimate of weak solutions to the equations just men-
tioned. One of the original traits of this short paper was a characterizationof the
boundary in terms of an isoperimetric function introduced by the author, which
enabledhimtostudythesharpdependenceoftheregularitypropertiesofsolutions
tothe Neumannproblemonthe behaviourofthe boundary.Adetailedexposition
of this work, containing a wealth of new ideas, was published in [18], 1969.
In[12],1963,Maz’yaobtainedhisfamousestimateofthecontinuitymodulus
of a solutionto the Dirichlet problemnear a boundary point, formulatedin terms
of the Wiener integral (see also [15], [16]). Later, a result of the same nature
was obtained by him for nonlinear equations including the p-Laplacian [19]. It
is noteworthy that the classical paper by Littman, Stampacchia and Weinberger
[8] on the Wiener regularity of a boundary point was translated into Russian by
Maz’ya for the Moscow collection of translations “Matematika” from a preprint,
even before its publication in a journal.
Of exceptional importance were Maz’ya’s counterexamples relating to the
19thand20thHilbertproblemsforhigher-orderellipticequationswhichappeared
in[17],1968,independentlyofandsimultaneouslywithanalogouscounterexamples
of E. De Giorgi and E. Giusti–M.Miranda.
The results of L. Cesari, R. Caccioppoli and especially E. De Giorgi on gen-
eralization of the notion of the surface area on nonsmooth surfaces played an
important role in the pioneering research of Maz’ya and his coauthors in the the-
oryofharmonicpotentialsonnonsmoothdomainsaswellasinthetheoryofspaces
of functions with bounded variation [1], [13], [2], [3].
The influence of G. Cimmino’s results of 1937 [4] on the Dirichlet problem
withboundarydatainL aswellasG.Fichera’sunifiedtheoryofelliptic-parabolic
p
equations [5] can be traced in Maz’ya’s breakthrough work on the generic degen-
erating oblique derivative problem [20].
Oneofthefundamentalresultsinthe theoryofpartialdifferentialequations,
theC.Miranda–Sh.Agmonmaximumprincipleforhigher-orderellipticequations,
was crucially developed by Maz’ya and his collaborators in several directions:
polyhedral domains [23], sharp constants [22], parabolic systems [7].
The above, by necessity a rather incomplete survey, clearly shows that the
ItalianschoolstimulatedtheearlyworkofMaz’yainspiteoftheironcurtain.With
time the contacts became bilateral and even personal. At the moment, Maz’ya is
collaborating with a number of Italian mathematicians which can be seen, for
instance, in some papers included into the present volume.