Cartesian currents in the calculus of variations: 2 Variational integrals
Gespeichert in:
| Beteiligte Personen: | , , |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Berlin [u.a.]
Springer
1998
|
| Schriftenreihe: | Ergebnisse der Mathematik und ihrer Grenzgebiete
3. Folge ; 38 |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008088428&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | XXIV, 697 S. Ill., graph. Darst. |
| ISBN: | 354064010X |
Internformat
MARC
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| 100 | 1 | |a Giaquinta, Mariano |d 1947- |e Verfasser |0 (DE-588)111595738 |4 aut | |
| 245 | 1 | 0 | |a Cartesian currents in the calculus of variations |n 2 |p Variational integrals |c Mariano Giaquinta ; Giuseppe Modica ; Jiři Souček |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1998 | |
| 300 | |a XXIV, 697 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Ergebnisse der Mathematik und ihrer Grenzgebiete : 3. Folge |v 38 | |
| 490 | 0 | |a Ergebnisse der Mathematik und ihrer Grenzgebiete : 3. Folge |v ... | |
| 700 | 1 | |a Modica, Giuseppe |d 1948- |e Verfasser |0 (DE-588)133455777 |4 aut | |
| 700 | 1 | |a Souček, Jiří |e Verfasser |4 aut | |
| 773 | 0 | 8 | |w (DE-604)BV011961231 |g 2 |
| 830 | 0 | |a Ergebnisse der Mathematik und ihrer Grenzgebiete |v 3. Folge ; 38 |w (DE-604)BV000899194 |9 38 | |
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| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-008088428 | |
Datensatz im Suchindex
| _version_ | 1819280370512691200 |
|---|---|
| adam_text | Contents
Volume II. Variational Integrals
1. Regular Variational Integrals 1
1 The Direct Methods 2
1.1 The Abstract Setting 3
(Lower semicontinuous and coercive functionals. Weierstrass theorem,
r relaxed functional)
1.2 Some Classical Lower Semicontinuity Theorems 10
(Lower semicontinuity with respect to the weak convergence in Ll. The
role of Banach Saks s theorem and Jensen s inequality. Regular and
smooth integrands in the Calculus of Variations)
1.3 A General Semicontinuity Theorem 19
(Lower semicontinuity with respect to the weak* convergence in L1)
2 Polyconvex Envelops and Regular Parametric Integrals 23
2.1 Poly convexity and Polyconvex Envelops 26
(A few facts from convex analysis, n vectors associated to the tangent
planes to graphs. Polyconvex functions and polyconvex envelopes)
2.2 Parametric Polyconvex Envelops of Integrands 37
(The parametric polyconvex l.s.c. envelop of an integrand. Mass and
comass with respect to the integrand /)
2.3 The Parametric Extension of Regular Integrals 44
(The parametric extension af an integral as integral of the parametric
polyconvex l.s.c. envelop)
2.4 The Polyconvex l.s.c. Extension of Some Lagrangians .... 45
(Area of graphs. The total variation of the gradient. The Dirichlet inte¬
gral. The p energy functional. The liquid crystal integrand)
3 Regular Integrals in the Class of Cartesian Currents 74
3.1 Parametric Integrands and Lower Semicontinuity 75
(Parametric integrands and lower semicontinuity of parametric integrals)
3.2 Existence of Minimizers in Classes of Cartesian Currents . 82
(Lower semicontinuity of the parametric extension of regular integrals.
Existence of minimizers in subclasses of Cartesian currents)
3.3 Relaxed Energies in the Setting of Cartesian Currents.... 88
(The relaxed functional in classes of Cartesian currents and maps)
3.4 Relaxed Energies in the Parametric Case 90
(The approximation problem for parametric integrals. A theorem of
Reshetnyak. Flat integrands and Federer s approximation theorem. In¬
teger multiplicity rectifiable and real minimizing currents)
4 Regular Integrals and Quasiconvexity 106
4.1 Quasiconvexity 107
(Quasiconvexity as necessary condition for semicontinuity. Rank one
convexity and Legendre Hadamard condition)
4.2 Quasiconvexity and Lower Semicontinuity 117
(Quasiconvexity as sufficient condition for semicontinuity in classes of
Sobolev maps or Cartesian currents)
xiv Contents Volume II. Variational Integrals
4.3 Ellipticity and Quasiconvexity 127
(Ellipticity, quasiconvexity and lower semicontinuity)
5 Notes 131
2. Finite Elasticity and Weak Diffeomorphisms 137
1 State Space and Stored Energies in Elasticity 139
1.1 Fields and Transformations 139
(Non local and non linear structure of transformations)
1.2 Kinematics 140
(Bodies, states, and deformations of a body. Deformations as 3 surfaces
in R6, or as graphs of diffeomorphisms)
1.3 Local Deformations 143
(Infinitesimal deformations as simple tangent vectors to the deformation
surface)
1.4 Perfectly Elastic Bodies: Stored Energy, Convexity and
Coercivity 147
(Stored energy: different forms and constitutive conditions. Polyconvex
ity and coercivity)
1.5 Variations and Stress 152
(Infinitesimal variations and the notion of stress. Piola Kirchhoff and
Cauchy stress tensors. Energy momentum tensor.)
2 Physical Implications on Kinematics and Stored Energies 155
2.1 Kinematical Principles in Elasticity: Weak Deformations . 156
(Material body and its parts. Impenetrability of matter. Weakly invert
ible maps. Weak one to one transformations. Existence of local defor¬
mations. Weak deformations. Elastic bodies and absence of fractures.
Elastic deformations)
2.2 Frame Indifference and Isotropy 169
(Frame indifference principle. Energies associated to isotropic materials)
2.3 Convexity like Conditions 170
(Convexity is not compatible with elasticity. Noll s condition. Polycon
vexity and Diff quasiconvexity)
2.4 Coercivity Conditions 175
(A discussion of the coercivity conditions)
2.5 Examples of Stored Energies 179
(Ogden type stored energies for isotropic materials)
3 Weak Diffeomorphisms 182
3.1 The Classes dif p 9(/?, Q) 183
(The class of (p, q) weak diffeomorphisms. Weak convergence. Closure
and compactness properties. An example of a discontinuous weak diffeo
morphism)
3.2 The Classes dif P (n, Sn) 191
(Weak diffeomorphisms with non prescribed range. Closure and com¬
pactness properties. Elastic deformations as weak diffeomorphisms)
3.3 Convergence Theorems for the Inverse Maps 199
(Convergence of the ranges and of the inverse maps)
3.4 General Weak Diffeomorphisms 204
(Weak diffeomorphisms with vertical and horizontal parts: structure,
closure, and compactness theorems)
3.5 The Dif classes 213
(The approximation problem for weak diffeomorphisms)
3.6 Volume Preserving Diffeomorphisms 214
(The Jacobian determinant of weak one to one maps and of weak de¬
formations)
4 Connectivity Properties of the Range of Weak Diffeomorphisms . 216
Contents Volume II. Variational Integrals xv
4.1 Connectivity of the Range of Sobolev Maps 217
(Connected sets and dn i connected sets. dn _ i connected sets are
mapped into connected sets by Sobolev maps)
4.2 Connectivity of the Range of Weak Diffeomorphisms 219
(Weak diffeomorphisms map connected sets into essentially connected
sets. Weak diffeomorphisms do not produce cavitation. Examples)
4.3 Regularity Properties of Locally Weak Invertible Maps ... 229
(Weak local diffeomorphisms. Local properties of weak local diffeomor¬
phisms. Vodopianov Goldstein s theorem. Courant Lebesgue lemma)
4.4 Global Invertibility of Weak Maps 238
(Conditions ensuring that weak local diffeomorphisms be one to one and
homeomorphisms)
4.5 An a.e. Open Map Theorem 243
(A.e. open sets and a.e. continuous maps. Weak diffeomorphisms in
W1 p, p n — 1, are a.e. open maps)
5 Composition 247
5.1 Composition of weak deformations 248
(Composition of one—to one maps and of weak diffeomorphisms)
5.2 On the Summability of Compositions 250
(Binet s formula and the summability of the composition)
5.3 Composition of Weak Diffeomorphisms 254
(The action of weak diffeomorphisms on Cartesian maps and the pseu
dogroup structure of weak diffeomorphisms. Weak convergence of com¬
positions)
6 Existence of Equilibrium Configurations 260
6.1 Existence Theorems 261
(The displacement pressure problem. Deformations with fractures)
6.2 Equilibrium and Conservation Equations 264
(Energy momentum conservation law and Cauchy s equilibrium equa¬
tion)
6.3 The Cavitation Problem 268
(Elastic deformations do not cavitate)
7 Notes 278
3. The Dirichlet Integral in Sobolev Spaces 281
1 Harmonic Maps Between Manifolds 281
1.1 First Variation and Inner Variations 283
(Euler variations and Euler Lagrange equation of the energy integral.
Inner variations, energy momentum tensor, inner and strong extremals.
Conformality relations. Stationary points. Parametric minimal surfaces)
1.2 Finding Harmonic Maps by Variational Methods 293
(Existence and the regularity problem. Mappings from Bn into the upper
hemisphere of S™. Mappings from B into S ^1)
2 Energy Minimizing Weak Harmonic Maps: Regularity Theory ... 296
2.1 Some Preliminaries. Reverse Holder Inequalities 297
(Some algebraic lemmas. The Dirichlet growth theorem of Morrey. Re¬
verse Holder inequalities with increasing supports)
2.2 Classical Regularity Results 303
(Morrey s regularity theorem for 2 dimensional weak harmonic maps)
2.3 An Optimal Regularity Theorem 307
(A partial regularity theorem and the existence and regularity of en¬
ergy minimizing harmonic maps with range in a regular ball: results by
Hildebrandt, Kaul, Widman, and Giaquinta, Giusti.)
2.4 The Partial Regularity Theorem 319
(The partial regularity theorem for energy minimizing weak harmonic
maps: Schoen Uhlenbeck result)
xvi Contents Volume II. Variational Integrals
3 Harmonic Maps in Homotopy Classes 333
3.1 The Action of W1 2 maps on Loops 334
(Courant Lebesgue lemma. In the two dimensional case the action on
loops is well denned for maps in W1 2)
3.2 Minimizing Energy with Homotopic Constraints 336
(Energy minimizing maps with prescribed action on loops. Schoen Yau,
Saks Uhlenbeck, Lemaire, Eells Sampson and Hamilton theorems)
3.3 Local Replacement by Harmonic Mappings: Bubbling .... 337
(Jost s replacement method and existence of minimal immersions of S2)
4 Weak and Stationary Harmonic Maps with Values into S2 339
4.1 The Partial Regularity Theory 339
(An alternative proof. More on the singular set)
4.2 Stationary Harmonic Maps 345
(Partial regularity results for stationary harmonic maps)
5 Notes 350
4. The Dirichlet Energy for Maps into S2 353
1 Variational Problems for Maps from a Domain of R2 into S2 ¦... 354
1.1 Harmonic Maps with Prescribed Degree 354
(Homotopic equivalent maps and degree. Bubbling off of spheres. The
stereographic and the modified stereographic projection, e—conformal
maps)
1.2 The Structure Theorem in cart2a(/2 x S2), fief2 362
(The structure and approximation theorems in cart2 1 (fi X S2))
1.3 Existence and Regularity of Minimizers 366
(The relaxed energy and existence of minimizers. Energy minimizing
maps with constant boundary value: Lemaire s theorem. The simplest
chiral model and instantons. Large solution for harmonic maps: Brezis
Coron and Jost result. A global regularity result)
2 Variational Problems from a Domain of R3 into S2 383
2.1 The Class cart2 1^ x S2), flcl3 385
(The £ field and homological singularities)
2.2 Density Results in W1 2(B3, S2) 392
(Approximation by maps which are smooth except at a discrete set of
points)
2.3 Dipoles and Gap Phenomenon 400
(Dipoles and the approximate dipoles. Lavrentiev or gap phenomenon)
2.4 The Structure Theorem in cart2 1 (O x S2), Q C R3 409
(Structure of the vertical part of Cartesian currents in cart21(J? X S2))
2.5 Approximation by Smooth Graphs: Dirichlet Data 412
(Weak approximation in energy by smooth graphs. The minimal con¬
nection and its continuity properties with respect to the W12 weak
convergence. Cart21(f2 X S2) = cart21(f? X S2). Weak approximability
by smooth maps in W^ 2(O X S2))
2.6 Approximation by Smooth Graphs: No Boundary Data... 419
(T belongs to cart21(/2 X S2) if and only if it can be approximated
weakly and in energy by smooth graphs Gu possibly with uk = «t on
an)
2.7 The Dirichlet Integral in cart2 ^/? x S2), ficl3 423
(The parametric polyconvex extension of the Dirichlet integral is its
relaxed or Lebesgue s extension. The relaxed of T (u, £2) in W12(i7, S2))
2.8 Minimizers of Variational Problems 429
(Variational problems and existence of minimizers)
Contents Volume II. Variational Integrals xvii
2.9 A Partial Regularity Result 433
(The absolutely continuous part ut of minimizers T is regular except on
a closed set whose Hausdorff dimension is not greater than 1. Tangent
cones)
2.10 The General Dipole Problem 449
(The coarea formula and the minimum energy of dipoles)
2.11 Singular Perturbations 452
(Trying to solve Dirichlet problem by approximating by singularly per¬
turbed functional of the type of Ginzburg Landau)
3 Notes 458
5. Some Regular and Non Regular Variational Problems 467
1 The Liquid Crystal Energy 467
1.1 The Sobolev Space Approach 470
(Existence and regularity of equilibrium configuration)
1.2 The Relaxed Energy 470
(Existence of equilibrium configurations for the relaxed energy. The
dipole problem. Relaxed energies in Sobolev spaces and Cartesian cur¬
rents. Equilibrium configurations with fractures)
1.3 The Dipole Problem 477
(Approximation in energy: irrotational and solenoidal dipoles. The gen¬
eral dipole problem)
2 The Dirichlet Integral in the Regular Case: Maps into S2 485
2.1 Maps with Values in S2 485
(Maps from a n dimensional space into S2 and the class cart2 (fix S2).
The (n 2) D field)
2.2 The Dipole Problem 489
(Degree with respect to a (n — 3)—curve and the dipole problem)
2.3 The Structure Theorem 494
(Structure theorem for currents in cart2 (fi x S2), Q C M )
3 The Dirichlet Integral in the Regular Case: Maps into a Manifold 496
3.1 The Class cart2 1{Q x y) 497
(The structure theorem for currents in cart2 (fi x y), Q C R2)
3.2 Spherical Vertical Parts and a Closure Theorem 501
(Reduced Cartesian currents. Closure theorem. Vertical parts of currents
in Cart21 (fi x y) axe of the type S2)
3.3 The Dirichlet Integral and Minimizers 506
4 The Dirichlet Integral in the Non Regular Case: a Homological
Theory 508
4.1 (n,p) Currents 509
((n.p) graphs and the classes t p . rectifiable (r, p) currents, (r,p)
mass, (r, p) boundary. Vertical (r, p) currents and cohomology. Integer
multiplicity rectifiable vector valued currents)
4.2 Graphs of Sobolev Maps 516
(Singularities of Sobolev maps and the currents P(u; r) and D(u;cr).
The class me W^ ((2, y))
4.3 p Dirichlet Graphs and Cartesian Currents 525
(The classes X ,, graph(fi x y), red P,, graph(fi x y), and cart (fix
y): closure theorems)
4.4 The Dirichlet Integral 534
(Representation. Minimizers and homological minimizers of the Dirichlet
integral)
4.5 Prescribing Homological Singularities 543
(s degree. Lower bounds for the dipole energy)
5 Notes 546
xviii Contents Volume II. Variational Integrals
6. The Non Parametric Area Functional 563
1 Area Minimizing Hypersurfaces 564
1.1 Parametric Surfaces of Least Area 564
(Hypersurfaces as Caccioppoli s boundaries: De Giorgi s regularity the¬
orem, monotonicity, Federer s regularity theorem. Surfaces as rectifiable
currents: Almgren s regularity theorem. Minimal surfaces as stationary
varifolds: a survey of Allard theory. Boundary regularity: Allard s and
Hardt Simon s results)
1.2 Non Parametric Minimal Surfaces of Codimension One ... 579
(Solvability of the Dirichlet problem. Bombieri De Giorgi Miranda a pri¬
ori estimate. The variational approach. Removable singularities. Liou
ville type theorems. Bernstein theorem. Bombieri De Giorgi Giusti the¬
orem on minimal cones)
2 Problems for Maps of Bounded Variation with Values in S1 590
2.1 Preliminaries 594
(Forms and currents in Q X S1. BV(n,R): a survey of results)
2.2 The Class cart(/2 x S1) 600
(The structure and the approximation theorems)
2.3 Relaxed Energies and Existence of Minimizers 610
(The area integral for maps into Sl: the relaxed area. Minimizers in
cart(J7 X S1). Dipole type problems)
3 Two Dimensional Minimal Surfaces 619
3.1 Plateau s Problem 619
(Morrey s e—conformality theorem. Douglas Rado existence theorem.
Hildebrandt s boundary regularity theorem. Branch points and embed¬
ded minimal surfaces: Fleming, Meeks Yau, and Chang results)
3.2 Existence of Two Dimensional Non Parametric Minimal
Surfaces 625
(Rado s theorem and existence, uniqueness, and regularity of two—di¬
mensional graphs of any codimension)
3.3 The Minimal Surface System 627
(Stationary graphs are not necessarily area minimizing. Existence and
non existence of stationary Lipschitz graphs. Isolated singularities are
not removable in high codimension. A Bernstein type result of Hilde
brandt, Jost, and Widman)
4 Least Area Mappings and Least Mass Currents 632
4.1 Topological Results 633
(Representation and homology of Lipschitz chains)
4.2 Main Results 635
(Least area mapping u : B —» R and least mass currents agree
if n 3. If n 3 the homotopy least area problem reduces to the
homology problem)
5 The Non parametric Area Integral 639
5.1 The Mass of Cartesian Currents and the Relaxed Area ... 641
(Graphs of finite mass which cannot be approximated in area by smooth
graphs)
5.2 Lebesgue s Area 649
(The mass of 2 dimensional continuous Cartesian maps is Lebesgue s
area of their graphs)
6 Notes 651
Bibliography 653
Index 683
Symbols 695
|
| any_adam_object | 1 |
| author | Giaquinta, Mariano 1947- Modica, Giuseppe 1948- Souček, Jiří |
| author_GND | (DE-588)111595738 (DE-588)133455777 |
| author_facet | Giaquinta, Mariano 1947- Modica, Giuseppe 1948- Souček, Jiří |
| author_role | aut aut aut |
| author_sort | Giaquinta, Mariano 1947- |
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| bvnumber | BV011961255 |
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| ctrlnum | (OCoLC)62032680 (DE-599)BVBBV011961255 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV011961255 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T10:21:46Z |
| institution | BVB |
| isbn | 354064010X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008088428 |
| oclc_num | 62032680 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-20 DE-29T DE-703 DE-824 DE-706 DE-634 DE-83 DE-11 DE-188 |
| owner_facet | DE-355 DE-BY-UBR DE-20 DE-29T DE-703 DE-824 DE-706 DE-634 DE-83 DE-11 DE-188 |
| physical | XXIV, 697 S. Ill., graph. Darst. |
| publishDate | 1998 |
| publishDateSearch | 1998 |
| publishDateSort | 1998 |
| publisher | Springer |
| record_format | marc |
| series | Ergebnisse der Mathematik und ihrer Grenzgebiete |
| series2 | Ergebnisse der Mathematik und ihrer Grenzgebiete : 3. Folge |
| spellingShingle | Giaquinta, Mariano 1947- Modica, Giuseppe 1948- Souček, Jiří Cartesian currents in the calculus of variations Ergebnisse der Mathematik und ihrer Grenzgebiete |
| title | Cartesian currents in the calculus of variations |
| title_auth | Cartesian currents in the calculus of variations |
| title_exact_search | Cartesian currents in the calculus of variations |
| title_full | Cartesian currents in the calculus of variations 2 Variational integrals Mariano Giaquinta ; Giuseppe Modica ; Jiři Souček |
| title_fullStr | Cartesian currents in the calculus of variations 2 Variational integrals Mariano Giaquinta ; Giuseppe Modica ; Jiři Souček |
| title_full_unstemmed | Cartesian currents in the calculus of variations 2 Variational integrals Mariano Giaquinta ; Giuseppe Modica ; Jiři Souček |
| title_short | Cartesian currents in the calculus of variations |
| title_sort | cartesian currents in the calculus of variations variational integrals |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008088428&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV011961231 (DE-604)BV000899194 |
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