Cartesian currents in the calculus of variations: 1 Cartesian currents
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
1998
|
| Series: | Ergebnisse der Mathematik und ihrer Grenzgebiete
3. Folge ; 37 |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Hier auch später erschienene, unveränderte Nachdrucke |
| Physical Description: | XXIV, 711 S. Ill., graph. Darst. |
| ISBN: | 3540640096 9783642083747 9783540640097 |
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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 1 |p Cartesian currents |c Mariano Giaquinta ; Giuseppe Modica ; Jiři Souček |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1998 | |
| 300 | |a XXIV, 711 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 37 | |
| 490 | 0 | |a Ergebnisse der Mathematik und ihrer Grenzgebiete : 3. Folge |v ... | |
| 500 | |a Hier auch später erschienene, unveränderte Nachdrucke | ||
| 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 1 |
| 830 | 0 | |a Ergebnisse der Mathematik und ihrer Grenzgebiete |v 3. Folge ; 37 |w (DE-604)BV000899194 |9 37 | |
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Record in the Search Index
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| adam_text | Contents
Volume I. Cartesian Currents
1. General Measure Theory 1
1 General Measure Theory 1
1.1 Measures and Integrals 1
(cr measures and outer measures, measurable sets, Caratheodory mea¬
sures. Measurable functions. Lebesgue s integral. Egoroff s, Beppo Levi s,
Fatou s and Lebesgue s theorems. Product measures and Fubini Tonelli
theorem)
1.2 Borel Regular and Radon Measures 10
(Borel functions and measures, Radon measures. Caratheodory s crite¬
rion in metric spaces. Vitali—Caratheodory theorem. Lusin s theorem)
1.3 Hausdorff Measures 12
(Hausdorff measures and Hausdorff dimension, spherical measures. Iso
diametric inequality. Cantor sets and Cantor Vitali functions. Caratheo¬
dory construction. Hausdorff measure of a product)
1.4 Lebesgue s, Radon Nikodym s and Riesz s Theorems 23
(Lebesgue s decomposition theorem, Radon—Nikodym differentiation the¬
orem. Vector valued measures and Riesz representation theorem)
1.5 Covering Theorems, Differentiation and Densities 29
(Vitali s and Besicovitch s covering theorems. Symmetric differentiation
and Radon Nikodym theorem. Densities. Approximate limits and mea
surability. Densities and Hausdorff measure)
2 Weak Convergence 36
2.1 Weak Convergence of Vector Valued Measures 36
(Definitions. Banach Steinhaus theorem and the compactness theorem.
Convergence as measures and L1 weak convergence. Lebesgue s theorem
about weak convergence in i1)
2.2 Typical Behaviours of Weakly Converging Sequences 40
(Oscillation, concentration, distribution, and concentration distribution.
Nonlinearity destroys the weak convergence)
2.3 Weak Convergence in Lq, q 1 44
(Weak convergence in Lp and weak and weak* convergence on Banach
spaces. Riemann Lebesgue s lemma. Radon Riesz theorem and variants)
2.4 Weak Convergence in L1 50
(The proof of Lebesgue theorem on weak convergence in L1 . Weak con¬
vergence of the product)
2.5 Concentration: Weak Convergence of Measures 55
(The universal character of the concentration distribution phenomenon.
The concentration compactness lemma)
2.6 Oscillations: Young Measures 58
(Equivalent definitions of Young measures. Examples)
2.7 More on Weak Convergence in L1 64
(Convergence in the sense of biting and convergence of the absolutely
continuous parts)
3 Notes 67
xiv Contents Volume I. Cartesian Currents
2. Integer Multiplicity Rectifiable Currents 69
1 Area and Coarea. Countably n Rectifiable Sets 69
1.1 Area and Coarea Formulas for Linear Maps 69
(Polar decomposition theorem. Area and change of variable formula for
linear maps. Coarea formula for linear maps. Cauchy Binet formula)
1.2 Area Formula for Lipschitz Maps 74
(Area formula for smooth and for Lipschitz maps: curves, graphs of
codimension one, parametric hypersurfaces, submanifolds, and graphs
of higher codimension)
1.3 Coarea Formula for Lipschitz Maps 82
(Coarea formula for smooth and for Lipschitz maps. C Sard type the¬
orem)
1.4 Rectifiable Sets and the Structure Theorem 90
(Countably n rectifiable and n rectifiable sets. The approximate tan¬
gent space of sets and measures. The rectifiability theorem for Radon
measures. Besicovitch Federer structure theorem)
1.5 The General Area and Coarea Formulas 99
(Area and coarea formulas for sets on manifolds and for rectifiable sets.
The divergence theorem)
2 Currents 103
2.1 Multivectors and Covectors 104
(fc vectors, exterior product of multivectors, simple fc vectors. Duality
between fc vectors and covectors. Inner product of multivectors. Sim¬
ple fc vectors and oriented fc planes. Simple n vectors in the Cartesian
product Rn X Rw. Characterization of simple n vectors in Mn x K .
Induced linear transformations)
2.2 Differential Forms 118
(Exterior differentiation, pullback, forms in a Cartesian product. Inte¬
gration of differential forms)
2.3 Currents: Basic Facts 122
(Currents and weak convergence of currents. Boundary and support of
currents. Product of currents. Currents with finite mass, currents which
are representable by integration. Lower semicontinuity of the mass.
Compactness closure theorem for currents with locally finite mass, n—
dimensional currents in Kn and BV functions. The constancy theorem.
Examples. Image of a current under a Lipschitz map. The homotopy
formula)
2.4 Integer Multiplicity Rectifiable Currents 136
(Currents carried by smooth graphs. Rectifiable and integer multiplic¬
ity rectifiable currents. The closure theorem for integer multiplicity 0
dimensional and 1 dimensional rectifiable currents. Examples. Image of
a rectifiable current under a Lipschitz map. The Cartesian product of
rectifiable currents)
2.5 Slicing 151
(Slices of codimension one. Slices of codimension larger than one)
2.6 The Deformation Theorem and Approximations 157
(The deformation theorem. Isoperimetric inequality. Weak and strong
polyhedral approximations. The strong approximation theorem for nor¬
mal currents)
2.7 The Closure Theorem 161
(The classical proof: slicing lemma, the boundary rectifiability theorem,
the rectifiability theorem. White s proof)
3 Notes 173
3. Cartesian Maps 175
1 Differentiability of Non Smooth Functions 179
Contents Volume I. Cartesian Currents xv
1.1 The Maximal Function and Lebesgue s Differentiation
Theorem 180
(The distribution function. Hardy Littlewood maximal function and
inequality. Lebesgue s differentiation theorem. Lebesgue s points. The
Hausdorff dimension of the set of non Lebesgue points. Calderon Zyg
mund decomposition argument. The class L log L)
1.2 Differentiability Properties of Whp Functions 192
(Differentiability in the V sense. Calderon Zygmund theorem. Morrey
Sobolev theorem)
1.3 Lusin Type Properties of W1 Functions 202
(Kirszbraun and Rademacher theorems. Lusin type theorems for Sobolev
functions. Whitney extension theorem. Liu s theorem)
1.4 Approximate Differential and Lusin Type Properties 210
(Approximate continuity and approximate differentiability. Lusin type
properties are equivalent to the approximate differentiability)
1.5 Area Formulas, Degree, and Graphs of Non Smooth Maps 218
(Area formula, graphs and degree for non smooth maps. Rado Reichel
derfer theorem)
2 Maps with Jacobian Minors in L1 228
2.1 The Class ^(fi.R^), Graphs and Boundaries 229
(The class ^41(^?,R ). The current integration over the graph. Conver¬
gence of graphs and minors)
2.2 Examples 233
(The map x/|i|. Homogeneous maps: boundary and degree)
2.3 Boundaries and Integration by Parts 238
(Approximate differential and distributional derivatives. Analytic for¬
mulas for boundaries. Boundary and pull back )
2.4 More on the Jacobian Determinant 247
(Maps in W1 1. The distributional determinant. Isoperimetric in¬
equality for the determinant. The class Ap q and 4,, i,n/(rt i). Higher
integrability of the determinant. BMO and Hardy space V}(R ))
2.5 Boundaries and Traces 265
(The boundary of a current integration over a graph and the trace in
the sense of Sobolev. Weak and strong anchorage)
3 Cartesian Maps 276
3.1 Weak Continuity of Minors 277
(Weak convergence of minors in L1, as measures, and convergence of
graphs. Examples. Reshetnyak s theorem)
3.2 The Class cart1^, RN): Closure and Compactness 285
(The class of Cartesian maps. The closure theorem. A compactness the¬
orem)
3.3 The Classes cartp(f2,R^), p 1 293
(The class cartp(4?,R ): closure and compactness theorems.)
4 Approximability of Cartesian Maps 296
4.1 The Transfinite Inductive Process 299
(Ordinal numbers. The transfinite inductive process and the weak se¬
quential closure of a set)
4.2 Weak and Strong Approximation of Minors 303
(Sequential weak closure and strong closure of smooth maps in the class
of Cartesian maps. Cart (fi, RN) = CART1 (Q, RN) C cart1 (f2, RN))
4.3 The Join of Cartesian Maps 313
(Composition and join of Cartesian maps. Weak continuity of the join)
5 Notes 318
xvi Contents Volume I. Cartesian Currents
4. Cartesian Currents in Euclidean Spaces 323
1 Functions of Bounded Variation 327
1.1 The Space BV{Q,R) 329
(The total variation. Semicontinuity. Approximation by smooth func¬
tions. A variational characterization of BV. Sobolev and Poincare in¬
equalities. Compactness. Fleming Rishel formula)
1.2 Caccioppoli Sets 340
(Sets of finite perimeter. Isoperimetric inequality. Approximation by
smooth open sets)
1.3 De Giorgi s Rectifiability Theorem 346
(Reduced boundary. De Giorgi s rectifiability theorem. Measure theo¬
retic boundary. Federer s characterization of Caccioppoli sets)
1.4 The Structure Theorem for BV Functions 354
(Jump points and jump sets. Regular and singular points. The struc¬
ture of the measure total variation. Lebesgue s points and approximate
differentiability of BV functions)
1.5 Subgraphs of BV Functions 371
(Characterization of BV functions in terms of their subgraphs; jump and
Cantor part of Du in terms of the reduced boundary of the subgraph of
u)
2 Cartesian Currents in Euclidean Spaces 379
2.1 Limit Currents of Smooth Graphs 380
(Toward the definition of Cartesian currents)
2.2 The Classes cart(J? x RN) and graph(/2 x RN) 384
(Cartesian currents and graph—currents)
2.3 The Structure Theorem 391
(The structure theorem. The map associated to a Cartesian current.
Weak convergence of Cartesian currents)
2.4 Cartesian Currents in Codimension One 403
(BV functions as Cartesian currents and representation formulas. Can
tor Vitali functions. SBV functions)
2.5 Examples of Cartesian Currents 411
(Bubbling off of circles, spheres, tori. Attaching cylinders. Examples of
vector valued BV functions. A Cartesian current with a Cantor mass
on minors. A Cartesian current which cannot be approximated weakly
by smooth graphs)
2.6 Radial Currents 439
(Currents associated to radial maps u(x) = U( x )x/ x . A closure the¬
orem)
3 Degree Theory 450
3.1 n Dimensional Currents and BV Functions 451
(Representation formula and decomposition theorem for n dimensional
currents in R . Constancy theorem and linear projections of normal
currents)
3.2 Degree Mapping and Degree of Cartesian Currents 460
(Definition and properties of the degree for Cartesian currents and maps)
3.3 The Degree of Continuous Maps 471
(The degree of continuous Cartesian maps agrees with the classical de¬
gree for continuous maps)
3.4 ^ Connected Components and the Degree 474
(Homologically connected components of a Caccioppoli set)
4 Notes 479
Contents Volume I. Cartesian Currents xvii
5. Cartesian Currents in Riemannian Manifolds 493
1 More About Currents 494
1.1 The Deformation Theorem 494
(Proof of Federer and Fleming deformation theorem and of the strong
approximation theorem)
1.2 Mollifying Currents 505
(e mollified of forms and currents. A representation formula for normal
currents)
1.3 Flat Chains 512
(Integral flat and flat chains and norms. Image of a flat chain. Fed
erer s flatness theorem. Mollification and a representation formula for
fiat chains. Federer s support theorem. Cochains)
2 Differential Forms and Cohomology 527
2.1 Forms on Manifolds 528
(Tangent and cotangent bundle. Null forms to a submanifold)
2.2 Hodge Operator 531
(Interior multiplication of vectors and covectors. Hodge operator. The
L2 inner product for forms)
2.3 Sobolev Spaces of Forms 536
(The classes L2p(X) and W* 2(X))
2.4 Harmonic Forms 538
(The codifferential 5. Laplace Beltrami operator on forms and the Dirich
let integral. Their expressions in local coordinates)
2.5 Hodge and Hodge Kodaira Morrey Theorems 543
(Gaffney Lemma. Hodge Kodaira Morrey decomposition theorem. De
Rham cohomology groups. Hodge representation theorem for cohomol¬
ogy classes)
2.6 Relative Cohomology: Hodge Morrey Decomposition 549
(Collar theorem. Tangential and normal part of a form. Coboundary op¬
erator. The lemma of Gaffney at the boundary. Hodge Morrey decom¬
position. Hodge representation theorem of relative cohomology classes)
2.7 Weitzenbock Formula 559
(Connections and covariant derivatives. Levi Civita connection. Second
•covariant derivatives. Curvature tensor and Laplace Beltrami operator
on forms)
2.8 Poincare and Poincare Lefschetz Dualities in Cohomology 565
(De Rham cohomology groups. Poincare duality. Relative cohomol¬
ogy groups on manifolds with boundary. Cohomology long sequence.
Poincare Lefschetz duality)
3 Currents and Real Homology of Compact Manifolds 570
3.1 Currents on Manifolds 572
(Currents in X. Flatness and constancy theorems)
3.2 Poincare and de Rham Dualities 574
(Isomorphism of (n — k) cohomology groups and fc homology groups.
Integration along fibers. Poincare dual form, de Rham duality between
cohomology and homology. Periods. Normal currents and classical real
homology)
3.3 Poincare Lefschetz and de Rham Dualities 589
(Relative homology. Homology long sequence. Poincare Lefschetz duality
theorem. De Rham theorem for manifolds with boundary. Relative real
homology classes are represented by minimal cycles)
3.4 Intersection of Currents and Kronecker Index 599
(Intersection of normal currents in R and on submanifolds of R . In¬
tersection of cycles is the wedge product of Poincare duals. Kronecker
index. Intersection index)
xviii Contents Volume I. Cartesian Currents
3.5 Relative Homology and Cohomology Groups 608
(Homology and cohomology in the Lipschitz category. Closure of cosets.
Generalized de Rham theorem)
4 Integral Homology 615
4.1 Integral Homology Groups 615
(Integral homology groups. Integral relative homology groups. Isoperi
metric inequalities and weak closure. Torsion groups. Integral and real
homology)
4.2 Intersection in Integral Homology 624
(Intersection of cycles on boundaryless manifolds. Intersection of cycles
on manifolds with boundary. Intersection index in integral homology. An
algebraic view of integral homology)
5 Maps Between Manifolds 631
5.1 Sobolev Classes of Maps Between Riemannian Manifolds . 632
(Density results of Schoen Uhlenbeck and Bethuel. d homotopy White s
results)
5.2 Cartesian Currents Between Manifolds 640
(Approximate differentiability. Area formula. Graphs. Cartesian cur¬
rents. The class cart21(i7 x y))
5.3 Homology Induced Maps: Manifolds Without Boundary .. 648
(Homology and cohomology maps associated to a Cartesian current)
5.4 Homology Induced Maps: Manifolds with Boundary 658
(Homology and cohomology maps associated to flat chains)
6 Notes 663
Bibliography 667
Index 697
Symbols 709
|
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| id | DE-604.BV011961241 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:19:14Z |
| institution | BVB |
| isbn | 3540640096 9783642083747 9783540640097 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008088417 |
| oclc_num | 313293647 |
| open_access_boolean | |
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| 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 DE-19 DE-BY-UBM |
| physical | XXIV, 711 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 |
| spelling | Giaquinta, Mariano 1947- Verfasser (DE-588)111595738 aut Cartesian currents in the calculus of variations 1 Cartesian currents Mariano Giaquinta ; Giuseppe Modica ; Jiři Souček Berlin [u.a.] Springer 1998 XXIV, 711 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Ergebnisse der Mathematik und ihrer Grenzgebiete : 3. Folge 37 Ergebnisse der Mathematik und ihrer Grenzgebiete : 3. Folge ... Hier auch später erschienene, unveränderte Nachdrucke Modica, Giuseppe 1948- Verfasser (DE-588)133455777 aut Souček, Jiří Verfasser aut (DE-604)BV011961231 1 Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge ; 37 (DE-604)BV000899194 37 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008088417&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| 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 1 Cartesian currents Mariano Giaquinta ; Giuseppe Modica ; Jiři Souček |
| title_fullStr | Cartesian currents in the calculus of variations 1 Cartesian currents Mariano Giaquinta ; Giuseppe Modica ; Jiři Souček |
| title_full_unstemmed | Cartesian currents in the calculus of variations 1 Cartesian currents Mariano Giaquinta ; Giuseppe Modica ; Jiři Souček |
| title_short | Cartesian currents in the calculus of variations |
| title_sort | cartesian currents in the calculus of variations cartesian currents |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008088417&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV011961231 (DE-604)BV000899194 |
| work_keys_str_mv | AT giaquintamariano cartesiancurrentsinthecalculusofvariations1 AT modicagiuseppe cartesiancurrentsinthecalculusofvariations1 AT soucekjiri cartesiancurrentsinthecalculusofvariations1 |