Fully Nonlinear PDEs in Real and Complex Geometry and Optics: Cetraro, Italy 2012, Editors: Cristian E. Gutiérrez, Ermanno Lanconelli

224 Pages·2014·1.631 MB·English

by  Luca Capogna, Pengfei Guan, Cristian E. Gutiérrez, Annamaria Montanari (auth.)

Lecture Notes in Mathematics 2087
CIME Foundation Subseries
Luca Capogna
Pengfei Guan
Cristian E. Gutiérrez
Annamaria Montanari
Fully Nonlinear
PDEs in Real and
Complex Geometry
and Optics
Cetraro, Italy 2012
Editors: Cristian E. Gutiérrez
Ermanno Lanconelli
Lecture Notes in Mathematics 2087
Editors:
J.-M.Morel,Cachan
B.Teissier,Paris
Forfurthervolumes:
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Luca Capogna Pengfei Guan
(cid:2)
Cristian E. Gutie´rrez Annamaria Montanari
(cid:2)
Fully Nonlinear PDEs in
Real and Complex Geometry
and Optics
Cetraro, Italy 2012
Editors:
Cristian E. Gutie´rrez
Ermanno Lanconelli
123
LucaCapogna PengfeiGuan
DepartmentofMathematicalSciences MathematicsandStatistics
WorcesterPolytechnicInstitute McGillUniversity
Worcester,MA,USA Montreal,QC,Canada
CristianE.Gutie´rrez AnnamariaMontanari
DepartmentofMathematics Dipto.Matematica
TempleUniversity Universita`diBologna
Philadelphia,PA,USA Bologna,Italy
ISBN978-3-319-00941-4 ISBN978-3-319-00942-1(eBook)
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Preface
This volume contains the notes from the CIME school “Fully Nonlinear PDEs in
RealandComplexGeometryandOptics”heldatCetraro(Cosenza,Italy)duringthe
weekofJuly9–13,2012.Theschoolconsistedoffourcourses:Extremalproblems
forquasiconformalmappingsinspacebyLucaCapogna,Fullynonlinearequations
ingeometrybyPengfeiGuan,Monge-Amper`etypeequationsandgeometricoptics
byCristian E. Gutie´rrez,andOn the LeviMonge–Ampe`reequationby Annamaria
Montanari.
Thepurposeoftheschoolwastopresentcurrentareasofresearch,arisingboth
in the theoretical and in the applied settings, that involve fully nonlinear partial
differentialequations.The solutionto these problemsrequiresthe developmentof
adhoctechniquesarisingintherelevantgeometricalframework,andincaseofthe
applications, determined by the physical laws governing the studied phenomena.
Precisely, the equations presented in the school come from conformal mapping
theory, differential geometry, optics, and geometric theory of several complex
variables.
Thefollowingisaquickoverviewofthecontentsofeachcourse.
LucaCapogna’slecturesprovidedanintroductiontotwovector-valuedextremal
L1-variational problems involving mappings. The first one concerns a classical
problem in geometric function theory that first arose in a work of Grotzsch from
1928:amongallorientationpreservingquasiconformalhomeomorphismswW(cid:2)!
(cid:2)0 whose traces agree with a (cid:2)given ma(cid:2)pping u0 W @(cid:2) ! @(cid:2)0, find one which
(cid:2) (cid:2)
minimizesthefunctionalu ! (cid:2)(cid:2) jduj (cid:2)(cid:2) . Variantsof thisproblemoccurwhen,
1
.detdu/n 1
instead of using boundary data, the class of competitors is defined in terms of a
fixedhomotopyclassorbyrequestingthatthetracesmapquasi-symmetrically@(cid:2)
into@(cid:2)0.Thesecondproblemgoesbacktotwopapersfrom1934,onebyWhitney
andtheotherbyMacShane,andleadingtotherecenttheoryofabsolutelyminimal
Lipschitz extensions. That is, let (cid:2) (cid:3) Rn be open, F (cid:3) (cid:2) be a compact set,
and let g 2 Lip.F;Rm/. Among all Lipschitz extensions of g from F to (cid:2), is
there a canonical unique extension that in some sense has the smallest possible
Lipschitznorm?Thenaturalquestionsarisinginconnectionwiththeseproblemsare
v
vi Preface
existence,uniqueness,and structure of the minimizers. The last few decadeshave
seen intense activity from different communities of mathematicians in the study
of both problems. However, at this time there does not seem to be much synergy
andcommunicationbetweenthesecommunities,bothintermsofsharedtechniques
used in the study of these problemsand in terms of common pointof views. One
ofthegoalsofCapogna’snotesistofostersuchsynergiesbyoutliningsomeofthe
commonfeaturesintheseproblems.
Pengfei Guan’s lectures considered nonlinear elliptic and parabolic partial dif-
ferentialequationsarisingfromgeometricproblemsforhypersurfacesinRnC1.The
notesareanintroductiontogeometricanalysis.Acurvature-typeellipticequationis
usedtosolvetheproblemofprescribingcurvaturemeasures,whichisaMinkowski-
typeproblem.CurvaturemeasuresaredefinedusingtheSteinerformula.Aninverse
meancurvature-typeparabolicequationisemployedfortheproofofisoperimetric-
typeinequalitiesforquermassintegralsofk-convexstar-shapeddomains.Bothtypes
of equations are fully nonlinear geometric PDEs. The emphasis of Guan’s notes
is on a priori estimates, a key step in the theory of fully nonlinear PDEs. The
presentationisself-containedandrequiresbasicknowledgeofPDEsandgeometry,
namelythestandardmaximumprinciplesforlinearellipticandparabolicequations,
the elementary formulas of Gauss, Codazzi, and Weingarten for hypersurfaces in
RnC1,andthecurvaturecommutatoridentities.Twobasicanddeepresultsareused
without proofs: the Evans–Krylov theorem for uniformly fully nonlinear elliptic
equationsandKrylov’stheoremforuniformlyparabolicfullynonlinearPDEs.
Gutie´rrez’scoursepresentedan introductionto Monge–Ampe`re-typeequations
and its applications to geometric optics. In general, these equations involve the
Jacobian determinant of a map and arise in the mathematical description of
numerousoptical,acoustic,andelectromagneticapplications,inparticular,in lens
and reflector antenna design. The geometric optics problems considered concern
refraction,reflection, or both.A typicalrefractionproblemthatwas consideredin
thelecturesisthefollowing:supposewehavetwohomogenousmediaI andIIwith
differentrefractiveindices, a lightbeam emanatesfrom a pointO, surroundedby
mediumI,andweseekaninterfacesurfaceseparatingmediaI andIIdescribedby
f(cid:3).x/x W x 2(cid:2)g;(cid:2)(cid:3) SnC1,andsuchthatallraysarerefractedintoeitherasetof
prescribeddirectionsorilluminateagivenobjectlyingonasurface,sayonaplane
inmediumII.Thefirsttypeofproblemiscalledfarfieldandthesecondnearfield.
Thesetwoproblemsareofdifferentmathematicalnature:thefirstoneisvariational
and the second is not. The input and outputintensities of radiation are prescribed
and so the problem of finding the interface surface is a typical inverse problem.
Gutie´rrez’s notes explain how to solve these problems: in the far field case using
masstransport,andinthenearfieldcaseusingtheMinkowskimethod.Thephysical
backgroundunderlyingtheseproblemsisexplainedusingMaxwellequations,and,
as a consequence, a deduction of Fresnel formulas is presented. These formulas
are finally applied to model the case when there is loss of energy due to internal
reflection.
Annamaria Montanari’s course focused on the Levi Monge–Ampe`re equation.
The equation is related to notions of curvatures associated with pseudoconvexity
Preface vii
andtheLeviform,inawaysimilartohowtheclassicalGaussandmeancurvatures
arerelatedtotheconvexityandtheHessianmatrix.Givenanonnegativefunctionk,
theLeviMonge–Ampe`reequationforthegraphofafunctionuWR2nC1 !Ris
detƒDk.xIu/.1CjDuj2/nC21;
where ƒ is the Levi form of the graph of u and Du is the Euclidean gradient
of u. More generally, in her notes, Montanari considers elementary symmetric
functions of the eigenvalues of the Levi form ƒ and shows that these curvature
equationscontaingeometricinformationabouthypersurfaceconsidered.Next,the
notes show that the curvature operator leads to a new class of second order fully
nonlinear equations whose characteristic form, when computed on generalized
pseudoconvex functions, is nonnegative definite with a kernel of dimension one.
Thus,theequationsarenotellipticatanypoint.However,theyhavethefollowing
redeeming feature: the missing ellipticity direction can be recovered by suitable
commutation relations. Using this property, existence, uniqueness, and regularity
of viscosity solutions of the Dirichlet problem for graphs with prescribed Levi
curvature are proved. Basic notions and results from the theory of functions of
several complex variables, and from the theory of viscosity solutions for fully
nonlineardegenerateellipticequations,arealsodescribedinthenotes.
Itisagreatpleasuretothankthespeakersfortheirveryinterestinglecturesand
for the useful lectures notes they have prepared for this volume. We also want to
thankalltheparticipantsoftheschoolfortheirinterestinthesesubjects.
Finally, we would like to warmly thankthe CIME foundationfor givingusthe
opportunity to organize and finance this school. We give our special gratitude to
PietroZecca,Elvira Mascolo,andall theCIME staff fortheirinvaluablehelpand
support.Also,aspecialthankstoMrs.UteMcCroryatSpringerforherassistance
inpreparingthisvolume.
Worcester,MA LucaCapogna
QC,Canada PengfeiGuan
Philadelphia,PA CristianE.Gutie´rrez
Bologna,Italy AnnamariaMontanari
Bologna,Italy ErmannoLanconelli
April5,2013

Contents
L1-ExtremalMappingsinAMLEandTeichmu¨llerTheory ............... 1
LucaCapogna
Curvature Measures, Isoperimetric Type Inequalities
andFullyNonlinearPDEs ...................................................... 47
PengfeiGuan
RefractionProblemsinGeometricOptics .................................... 95
CristianE.Gutie´rrez
OntheLeviMonge-Ampe`reEquation......................................... 151
AnnamariaMontanari
ListofParticipants............................................................... 209
ix