Table Of ContentLecture Notes of the Unione Matematica Italiana
Pascal Cherrier
Albert Milani
Evolution
Equations of
von Karman
Type
Lecture Notes of 17
the Unione Matematica Italiana
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Pascal Cherrier (cid:129) Albert Milani
Evolution Equations
of von Karman Type
123
PascalCherrier AlbertMilani
DépartmentdeMathématiques DepartmentofMathematics
UniversitéPierreetMarieCurie UniversityofWisconsin
Paris,France Milwaukee,WI,USA
ISSN1862-9113 ISSN1862-9121 (electronic)
LectureNotesoftheUnioneMatematicaItaliana
ISBN978-3-319-20996-8 ISBN978-3-319-20997-5 (eBook)
DOI10.1007/978-3-319-20997-5
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Preface
In these notes, we consider two kinds of nonlinear evolution problems of von
Karman type on R2m, m (cid:2) 2. Each of these problems consists of a system that
resultsfromthecouplingoftwohighlynonlinearpartialdifferentialequations,one
hyperbolic or parabolic, and the other elliptic. These systems are called “of von
Karman type” because of a formal analogy with the well-known equationsof the
samenameinthetheoryofelasticityinR2.
1 TheClassicalEquations
1. To describe the classical hyperbolic von Karman system in R2, we introduce
the nonlinear coupling of the second order derivatives of two sufficiently smooth
functionsgDg.x;y/andhDh.x;y/,definedby
(cid:2) (cid:3)
g g
Œg;h(cid:2)WDdet xx xy ; (1)
h h
yx yy
andthenweset
N.g;h/WDŒg;h(cid:2)CŒh;g(cid:2)Dg h Cg h (cid:3)2g h : (2)
xx yy yy xx xy xy
TheclassicalvonKarmanequationsinR2consistofthesystem
u C(cid:3)2uDN.f;u/CN.';u/; (3)
tt
(cid:3)2f D(cid:3)N.u;u/; (4)
where (cid:3) the usual Laplace operator in R2, and ' D '.t;x;y/ is a given external
source. Equations (3) and (4) model the dynamics of the vertical oscillations
(buckling) of an elastic two-dimensional thin plate, represented by a bounded
vii
viii Preface
domain(cid:4) (cid:4) R2,duetobothinternalandexternalstresses.Moreprecisely,inthis
modeltheunknownfunctionuDu.t;x;y/isameasureoftheverticaldisplacement
of the plate; Eq.(4) formally defines a map u 7! f.u/, where f.u/ represents the
so-calledAiry stress function,which is related to the internalelastic forcesacting
on the plate, and depends on its deformationu; finally, ' represents the action of
the external stress forces. Typically, Eqs.(3)C(4) are supplemented by the initial
conditions
u.0/Du0; ut.0/Du1; (5)
whereu0andu1areagiveninitialconfigurationofthedisplacementanditsvelocity,
andbyappropriateconstraintsonuattheboundaryof(cid:4).
2.Adetailedandprecisedescriptionofthemodelingissuesrelatedtotheclassical
vonKarmanequations,andadiscussionoftheirphysicalmotivations,canbefound
in, e.g.,Ciarlet andRabier [12], or in Ciarlet [10,11];in addition,we refer to the
recent, exhaustive study by Chuesov and Lasiecka [9] of a large class of initial-
boundaryvalueproblemsofvonKarmantypeondomainsofR2,withamultitude
ofdifferentboundaryconditions,includingnonlinearones.Thestationarystate of
theclassicalvonKarmanequations,describedbythenonlinearellipticsystem
(cid:3)2u DN.f;u/CN.';u/; (6)
(cid:3)2f D(cid:3)N.u;u/; (7)
has been investigated by several authors; in particular, Berger [3], devised a
remarkable variational method to establish the existence of suitably regular solu-
tions to the stationary system (6)C(7) in a bounded domain of R2, subject to
appropriate boundary conditions. Weak solutions of the corresponding system of
evolution (3)C(4)C(5), again under appropriate boundary conditions, have been
established,amongothers,byLions[21,Chap.1,Sect.4],andFavinietal.[15,16],
andChuesovandLasiecka[9].
2 TheGeneralized Equations
1. To introduce the generalization of the von Karman system (3)C(4) we wish to
consider,we nowletm 2 N(cid:2)2, and,givenmC1 smoothfunctionsu1; ::: ;um; u
definedonR2m,weset
N.u1; ::: ;um/ WD ıji11(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)jimmrij11u1 (cid:5)(cid:5)(cid:5) rijmmum; (8)
M.u/ WD N.u; ::: ;u/DmŠ(cid:5) .r2u/; (9)
m
2 TheGeneralizedEquations ix
wherewe adoptthe usualsummationconventionforrepeatedindices,anduse the
following notations. For i1; ::: ;im, j1; ::: ;jm 2 f1; ::: 2mg, ıji11(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)jimm denotes the
Kronecker tensor; for 1 (cid:6) i; j (cid:6) 2m, rj WD @@, and (cid:5) is the m-th elementary
i i j m
symmetricfunctionoftheeigenvalues(cid:6) D(cid:6) .@@u/,1(cid:6)k (cid:6)2m,oftheHessian
k k i j
matrixH.u/WDŒ@@u(cid:2),thatis,
i j
X
(cid:5) .r2u/WD (cid:6) (cid:5)(cid:5)(cid:5) (cid:6) : (10)
m k1 km
1(cid:4)k1<k2<:::<km(cid:4)2m
Wealsointroducetheconvention
(cid:4) (cid:5)
N u.1k1/; ::: ;u.pkp/ WDN.„u1; :ƒ:‚: ;u…1; ::: „up; :ƒ:‚: ;u…p/; (11)
k1factors kpfactors
withk1C (cid:5)(cid:5)(cid:5) Ckp Dm,andset(cid:3)WD(cid:3)rjju.
InLemma1.3.1ofChap.1,weshallshowthattheellipticequation
(cid:3)mf D(cid:3)M.u/ (12)
canbeuniquelysolved,in asuitablefunctionalframe,forf intermsofu,thereby
defining a map u 7! f WD f.u/. Let T > 0. Given a source term ' defined on
Œ0;T(cid:2) (cid:7) R2m, we consider the Cauchy problem, of hyperbolic type, in which we
wishtodetermineafunctionuonŒ0;T(cid:2)(cid:7)R2m,satisfyingtheequation
u C(cid:3)muDN.f.u/;u.m(cid:5)1//CN.'.m(cid:5)1/;u/; (13)
tt
andsubjecttotheinitialconditions(5), where,now,u0 andu1 aregivenfunctions
definedonR2m.WerefertothisCauchyproblem,thatis,explicitly,to(13)C(12)C
(5),as“problem(VKH)”.
Problem(VKH)appearstobeanalogoustotheoriginalvonKarmansystem(3)
and(4)onR2,butthisanalogyisonlyformal,inthefollowingsense.Letddenote
the space dimension. In the linear part at the left side of Eqs.(3) and (4) of the
original system, the order of the differential operator (cid:3)2 is twice the dimension
of space (i.e., 4 D 2d, d D 2), and the nonlinear operators of Monge-Ampère
typeattherightsideoftheequationsaredefinedintermsofthecompleteHessian
of functions depending on u, f, and '. In contrast, at the left side of Eqs.(13)
and(12)theorderofthedifferentialoperator(cid:3)mequalsthedimensionofspace(i.e.,
2m D d), while the Monge-Ampèreoperatorsat the right side of these equations
aredefinedintermsofelementarysymmetricfunctionsoforderm D d ofHessian
2
matricesoffunctionsdependingonu,f,and'.Toillustratethisdifferenceexplicitly,
intheoriginalequation(4)thetermN.u;u/istwicethedeterminantoftheHessian
