Direct methods in the calculus of variations /:
A comprehensive discussion on the existence and regularity of minima of regular integrals in the calculus of variations and of solutions to elliptic partial differential equations and systems of the second order. While direct methods for the existence of solutions are well-known and were widely-used...
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| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
River Edge, NJ :
World Scientific,
©2003.
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| Online-Zugang: | Volltext |
| Zusammenfassung: | A comprehensive discussion on the existence and regularity of minima of regular integrals in the calculus of variations and of solutions to elliptic partial differential equations and systems of the second order. While direct methods for the existence of solutions are well-known and were widely-used in the 20th century, the regularity of the minima was always obtained by means of the Euler equation as a part of the general theory of partial differential equations. In this work, using the notion of the quasi-minimum introduced by Giaquinta and the author, the direct methods are extended to the regularity of the minima of functionals in the calculus of variations, and of solutions to partial differential equations. This unified treatment offers a substantial economy in the assumptions, and permits a deeper understanding of the nature of the regularity and singularities of the solutions. The volume is essentially self-contained, and requires only a general knowledge of the elements of Lebesgue integration theory. |
| Beschreibung: | 1 online resource (vii, 403 pages) |
| Bibliographie: | Includes bibliographical references (pages 377-398) and index. |
| ISBN: | 9789812795557 9812795553 1281935905 9781281935908 |
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| 100 | 1 | |a Giusti, Enrico. |0 http://id.loc.gov/authorities/names/n80072414 | |
| 245 | 1 | 0 | |a Direct methods in the calculus of variations / |c Enrico Giusti. |
| 246 | 3 | 0 | |a Calculus of variations |
| 260 | |a River Edge, NJ : |b World Scientific, |c ©2003. | ||
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| 504 | |a Includes bibliographical references (pages 377-398) and index. | ||
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| 520 | |a A comprehensive discussion on the existence and regularity of minima of regular integrals in the calculus of variations and of solutions to elliptic partial differential equations and systems of the second order. While direct methods for the existence of solutions are well-known and were widely-used in the 20th century, the regularity of the minima was always obtained by means of the Euler equation as a part of the general theory of partial differential equations. In this work, using the notion of the quasi-minimum introduced by Giaquinta and the author, the direct methods are extended to the regularity of the minima of functionals in the calculus of variations, and of solutions to partial differential equations. This unified treatment offers a substantial economy in the assumptions, and permits a deeper understanding of the nature of the regularity and singularities of the solutions. The volume is essentially self-contained, and requires only a general knowledge of the elements of Lebesgue integration theory. | ||
| 505 | 0 | |a Introduction -- ch. 1. Semi-classical theory. 1.1. The maximum principle. 1.2. The bounded slope condition. 1.3. Barriers. 1.4. The area functional. 1.5. Non-existence of minimal surfaces. 1.6. Notes and comments -- ch. 2. Measurable functions. 2.1. L[symbol] spaces. 2.2. Test functions and mollifiers. 2.3. Morrey's and Campanato's spaces. 2.4. The lemmas of John and Nirenberg. 2.5. Interpolation. 2.6. The Hausdorff measure. 2.7. Notes and comments -- ch. 3. Sobolev spaces. 3.1. Partitions of unity. 3.2. Weak derivatives. 3.3. The Sobolev spaces W[symbol]. 3.4. Imbedding theorems. 3.5. Compactness. 3.6. Inequalities. 3.7. Traces. 3.8. The values of W[symbol] functions. 3.9. Notes and comments -- ch. 4. Convexity and semicontinuity. 4.1. Preliminaries. 4.2. Convex functional. 4.3. Semicontinuity. 4.4. An existence theorem. 4.5. Notes and comments -- ch. 5. Quasi-convex functional. 5.1. Necessary conditions. 5.2. First semicontinuity results. 5.3. The Quasi-convex envelope. 5.4. The Ekeland variational principle. 5.5. Semicontinuity. 5.6. Coerciveness and existence. 5.7. Notes and comments -- ch. 6. Quasi-minima. 6.1. Preliminaries. 6.2. Quasi-minima and differential quations. 6.3. Cubical quasi-minima. 6.4. L[symbol] estimates for the gradient. 6.5. Boundary estimates. 6.6. Notes and comments -- ch. 7. Hölder continuity. 7.1. Caccioppoli's inequality. 7.2. De Giorgi classes. 7.3. Quasi-minima. 7.4. Boundary regularity. 7.5. The Harnack inequality. 7.6. The homogeneous case. 7.7. w-minima. 7.8. Boundary regularity. 7.9. Notes and comments -- ch. 8. First derivatives. 8.1. The difference quotients. 8.2. Second derivatives. 8.3. Gradient estimates. 8.4. Boundary estimates. 8.5. w-minima. 8.6. Hölder continuity of the derivatives (p = 2). 8.7. Other gradient estimates. 8.8. Hölder continuity of the derivatives (p ≠ 2). 8.9. Elliptic equations. 8.10. Notes and comments -- ch. 9. Partial regularity. 9.1. Preliminaries. 9.2. Quadratic functionals. 9.3. The second Caccioppoli inequality. 9.4. The case F = F(z) (p = 2). 9.5. Partial regularity. 9.6. Notes and Comments -- ch. 10. Higher derivatives. 10.1. Hilbert regularity. 10.2. Constant coefficients. 10.3. Continuous coefficients. 10.4. L[symbol] estimates. 10.5. Minima of functionals. 10.6. Notes and comments. | |
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Datensatz im Suchindex
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| adam_text | |
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| author | Giusti, Enrico |
| author_GND | http://id.loc.gov/authorities/names/n80072414 |
| author_facet | Giusti, Enrico |
| author_role | |
| author_sort | Giusti, Enrico |
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| contents | Introduction -- ch. 1. Semi-classical theory. 1.1. The maximum principle. 1.2. The bounded slope condition. 1.3. Barriers. 1.4. The area functional. 1.5. Non-existence of minimal surfaces. 1.6. Notes and comments -- ch. 2. Measurable functions. 2.1. L[symbol] spaces. 2.2. Test functions and mollifiers. 2.3. Morrey's and Campanato's spaces. 2.4. The lemmas of John and Nirenberg. 2.5. Interpolation. 2.6. The Hausdorff measure. 2.7. Notes and comments -- ch. 3. Sobolev spaces. 3.1. Partitions of unity. 3.2. Weak derivatives. 3.3. The Sobolev spaces W[symbol]. 3.4. Imbedding theorems. 3.5. Compactness. 3.6. Inequalities. 3.7. Traces. 3.8. The values of W[symbol] functions. 3.9. Notes and comments -- ch. 4. Convexity and semicontinuity. 4.1. Preliminaries. 4.2. Convex functional. 4.3. Semicontinuity. 4.4. An existence theorem. 4.5. Notes and comments -- ch. 5. Quasi-convex functional. 5.1. Necessary conditions. 5.2. First semicontinuity results. 5.3. The Quasi-convex envelope. 5.4. The Ekeland variational principle. 5.5. Semicontinuity. 5.6. Coerciveness and existence. 5.7. Notes and comments -- ch. 6. Quasi-minima. 6.1. Preliminaries. 6.2. Quasi-minima and differential quations. 6.3. Cubical quasi-minima. 6.4. L[symbol] estimates for the gradient. 6.5. Boundary estimates. 6.6. Notes and comments -- ch. 7. Hölder continuity. 7.1. Caccioppoli's inequality. 7.2. De Giorgi classes. 7.3. Quasi-minima. 7.4. Boundary regularity. 7.5. The Harnack inequality. 7.6. The homogeneous case. 7.7. w-minima. 7.8. Boundary regularity. 7.9. Notes and comments -- ch. 8. First derivatives. 8.1. The difference quotients. 8.2. Second derivatives. 8.3. Gradient estimates. 8.4. Boundary estimates. 8.5. w-minima. 8.6. Hölder continuity of the derivatives (p = 2). 8.7. Other gradient estimates. 8.8. Hölder continuity of the derivatives (p ≠ 2). 8.9. Elliptic equations. 8.10. Notes and comments -- ch. 9. Partial regularity. 9.1. Preliminaries. 9.2. Quadratic functionals. 9.3. The second Caccioppoli inequality. 9.4. The case F = F(z) (p = 2). 9.5. Partial regularity. 9.6. Notes and Comments -- ch. 10. Higher derivatives. 10.1. Hilbert regularity. 10.2. Constant coefficients. 10.3. Continuous coefficients. 10.4. L[symbol] estimates. 10.5. Minima of functionals. 10.6. Notes and comments. |
| ctrlnum | (OCoLC)263159070 |
| dewey-full | 515/.64 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.64 |
| dewey-search | 515/.64 |
| dewey-sort | 3515 264 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Electronic eBook |
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While direct methods for the existence of solutions are well-known and were widely-used in the 20th century, the regularity of the minima was always obtained by means of the Euler equation as a part of the general theory of partial differential equations. In this work, using the notion of the quasi-minimum introduced by Giaquinta and the author, the direct methods are extended to the regularity of the minima of functionals in the calculus of variations, and of solutions to partial differential equations. This unified treatment offers a substantial economy in the assumptions, and permits a deeper understanding of the nature of the regularity and singularities of the solutions. The volume is essentially self-contained, and requires only a general knowledge of the elements of Lebesgue integration theory.</subfield></datafield><datafield tag="505" ind1="0" ind2=" "><subfield code="a">Introduction -- ch. 1. Semi-classical theory. 1.1. The maximum principle. 1.2. The bounded slope condition. 1.3. Barriers. 1.4. The area functional. 1.5. Non-existence of minimal surfaces. 1.6. Notes and comments -- ch. 2. Measurable functions. 2.1. L[symbol] spaces. 2.2. Test functions and mollifiers. 2.3. Morrey's and Campanato's spaces. 2.4. The lemmas of John and Nirenberg. 2.5. Interpolation. 2.6. The Hausdorff measure. 2.7. Notes and comments -- ch. 3. Sobolev spaces. 3.1. Partitions of unity. 3.2. Weak derivatives. 3.3. The Sobolev spaces W[symbol]. 3.4. Imbedding theorems. 3.5. Compactness. 3.6. Inequalities. 3.7. Traces. 3.8. The values of W[symbol] functions. 3.9. Notes and comments -- ch. 4. Convexity and semicontinuity. 4.1. Preliminaries. 4.2. Convex functional. 4.3. Semicontinuity. 4.4. An existence theorem. 4.5. Notes and comments -- ch. 5. Quasi-convex functional. 5.1. Necessary conditions. 5.2. First semicontinuity results. 5.3. The Quasi-convex envelope. 5.4. The Ekeland variational principle. 5.5. Semicontinuity. 5.6. Coerciveness and existence. 5.7. Notes and comments -- ch. 6. Quasi-minima. 6.1. Preliminaries. 6.2. Quasi-minima and differential quations. 6.3. Cubical quasi-minima. 6.4. L[symbol] estimates for the gradient. 6.5. Boundary estimates. 6.6. Notes and comments -- ch. 7. Hölder continuity. 7.1. Caccioppoli's inequality. 7.2. De Giorgi classes. 7.3. Quasi-minima. 7.4. Boundary regularity. 7.5. The Harnack inequality. 7.6. The homogeneous case. 7.7. w-minima. 7.8. Boundary regularity. 7.9. Notes and comments -- ch. 8. First derivatives. 8.1. The difference quotients. 8.2. Second derivatives. 8.3. Gradient estimates. 8.4. Boundary estimates. 8.5. w-minima. 8.6. Hölder continuity of the derivatives (p = 2). 8.7. Other gradient estimates. 8.8. Hölder continuity of the derivatives (p ≠ 2). 8.9. Elliptic equations. 8.10. Notes and comments -- ch. 9. Partial regularity. 9.1. Preliminaries. 9.2. Quadratic functionals. 9.3. The second Caccioppoli inequality. 9.4. The case F = F(z) (p = 2). 9.5. Partial regularity. 9.6. Notes and Comments -- ch. 10. Higher derivatives. 10.1. Hilbert regularity. 10.2. Constant coefficients. 10.3. Continuous coefficients. 10.4. L[symbol] estimates. 10.5. Minima of functionals. 10.6. 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| indexdate | 2024-10-25T16:16:55Z |
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| publisher | World Scientific, |
| record_format | marc |
| spelling | Giusti, Enrico. http://id.loc.gov/authorities/names/n80072414 Direct methods in the calculus of variations / Enrico Giusti. Calculus of variations River Edge, NJ : World Scientific, ©2003. 1 online resource (vii, 403 pages) text txt rdacontent computer c rdamedia online resource cr rdacarrier Includes bibliographical references (pages 377-398) and index. Print version record. A comprehensive discussion on the existence and regularity of minima of regular integrals in the calculus of variations and of solutions to elliptic partial differential equations and systems of the second order. While direct methods for the existence of solutions are well-known and were widely-used in the 20th century, the regularity of the minima was always obtained by means of the Euler equation as a part of the general theory of partial differential equations. In this work, using the notion of the quasi-minimum introduced by Giaquinta and the author, the direct methods are extended to the regularity of the minima of functionals in the calculus of variations, and of solutions to partial differential equations. This unified treatment offers a substantial economy in the assumptions, and permits a deeper understanding of the nature of the regularity and singularities of the solutions. The volume is essentially self-contained, and requires only a general knowledge of the elements of Lebesgue integration theory. Introduction -- ch. 1. Semi-classical theory. 1.1. The maximum principle. 1.2. The bounded slope condition. 1.3. Barriers. 1.4. The area functional. 1.5. Non-existence of minimal surfaces. 1.6. Notes and comments -- ch. 2. Measurable functions. 2.1. L[symbol] spaces. 2.2. Test functions and mollifiers. 2.3. Morrey's and Campanato's spaces. 2.4. The lemmas of John and Nirenberg. 2.5. Interpolation. 2.6. The Hausdorff measure. 2.7. Notes and comments -- ch. 3. Sobolev spaces. 3.1. Partitions of unity. 3.2. Weak derivatives. 3.3. The Sobolev spaces W[symbol]. 3.4. Imbedding theorems. 3.5. Compactness. 3.6. Inequalities. 3.7. Traces. 3.8. The values of W[symbol] functions. 3.9. Notes and comments -- ch. 4. Convexity and semicontinuity. 4.1. Preliminaries. 4.2. Convex functional. 4.3. Semicontinuity. 4.4. An existence theorem. 4.5. Notes and comments -- ch. 5. Quasi-convex functional. 5.1. Necessary conditions. 5.2. First semicontinuity results. 5.3. The Quasi-convex envelope. 5.4. The Ekeland variational principle. 5.5. Semicontinuity. 5.6. Coerciveness and existence. 5.7. Notes and comments -- ch. 6. Quasi-minima. 6.1. Preliminaries. 6.2. Quasi-minima and differential quations. 6.3. Cubical quasi-minima. 6.4. L[symbol] estimates for the gradient. 6.5. Boundary estimates. 6.6. Notes and comments -- ch. 7. Hölder continuity. 7.1. Caccioppoli's inequality. 7.2. De Giorgi classes. 7.3. Quasi-minima. 7.4. Boundary regularity. 7.5. The Harnack inequality. 7.6. The homogeneous case. 7.7. w-minima. 7.8. Boundary regularity. 7.9. Notes and comments -- ch. 8. First derivatives. 8.1. The difference quotients. 8.2. Second derivatives. 8.3. Gradient estimates. 8.4. Boundary estimates. 8.5. w-minima. 8.6. Hölder continuity of the derivatives (p = 2). 8.7. Other gradient estimates. 8.8. Hölder continuity of the derivatives (p ≠ 2). 8.9. Elliptic equations. 8.10. Notes and comments -- ch. 9. Partial regularity. 9.1. Preliminaries. 9.2. Quadratic functionals. 9.3. The second Caccioppoli inequality. 9.4. The case F = F(z) (p = 2). 9.5. Partial regularity. 9.6. Notes and Comments -- ch. 10. Higher derivatives. 10.1. Hilbert regularity. 10.2. Constant coefficients. 10.3. Continuous coefficients. 10.4. L[symbol] estimates. 10.5. Minima of functionals. 10.6. Notes and comments. Calculus of variations. http://id.loc.gov/authorities/subjects/sh85018809 Calcul des variations. MATHEMATICS Calculus. bisacsh MATHEMATICS Mathematical Analysis. bisacsh Calculus of variations fast has work: Direct methods in the calculus of variations (Text) https://id.oclc.org/worldcat/entity/E39PCFG3CtvPhxmxvPMpYPV6Kd https://id.oclc.org/worldcat/ontology/hasWork Print version: Giusti, Enrico. Direct methods in the calculus of variations. River Edge, NJ : World Scientific, ©2003 9812380434 9789812380432 (DLC) 2005277642 (OCoLC)61278752 FWS01 ZDB-4-EBA FWS_PDA_EBA https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=235657 Volltext CBO01 ZDB-4-EBA FWS_PDA_EBA https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=235657 Volltext |
| spellingShingle | Giusti, Enrico Direct methods in the calculus of variations / Introduction -- ch. 1. Semi-classical theory. 1.1. The maximum principle. 1.2. The bounded slope condition. 1.3. Barriers. 1.4. The area functional. 1.5. Non-existence of minimal surfaces. 1.6. Notes and comments -- ch. 2. Measurable functions. 2.1. L[symbol] spaces. 2.2. Test functions and mollifiers. 2.3. Morrey's and Campanato's spaces. 2.4. The lemmas of John and Nirenberg. 2.5. Interpolation. 2.6. The Hausdorff measure. 2.7. Notes and comments -- ch. 3. Sobolev spaces. 3.1. Partitions of unity. 3.2. Weak derivatives. 3.3. The Sobolev spaces W[symbol]. 3.4. Imbedding theorems. 3.5. Compactness. 3.6. Inequalities. 3.7. Traces. 3.8. The values of W[symbol] functions. 3.9. Notes and comments -- ch. 4. Convexity and semicontinuity. 4.1. Preliminaries. 4.2. Convex functional. 4.3. Semicontinuity. 4.4. An existence theorem. 4.5. Notes and comments -- ch. 5. Quasi-convex functional. 5.1. Necessary conditions. 5.2. First semicontinuity results. 5.3. The Quasi-convex envelope. 5.4. The Ekeland variational principle. 5.5. Semicontinuity. 5.6. Coerciveness and existence. 5.7. Notes and comments -- ch. 6. Quasi-minima. 6.1. Preliminaries. 6.2. Quasi-minima and differential quations. 6.3. Cubical quasi-minima. 6.4. L[symbol] estimates for the gradient. 6.5. Boundary estimates. 6.6. Notes and comments -- ch. 7. Hölder continuity. 7.1. Caccioppoli's inequality. 7.2. De Giorgi classes. 7.3. Quasi-minima. 7.4. Boundary regularity. 7.5. The Harnack inequality. 7.6. The homogeneous case. 7.7. w-minima. 7.8. Boundary regularity. 7.9. Notes and comments -- ch. 8. First derivatives. 8.1. The difference quotients. 8.2. Second derivatives. 8.3. Gradient estimates. 8.4. Boundary estimates. 8.5. w-minima. 8.6. Hölder continuity of the derivatives (p = 2). 8.7. Other gradient estimates. 8.8. Hölder continuity of the derivatives (p ≠ 2). 8.9. Elliptic equations. 8.10. Notes and comments -- ch. 9. Partial regularity. 9.1. Preliminaries. 9.2. Quadratic functionals. 9.3. The second Caccioppoli inequality. 9.4. The case F = F(z) (p = 2). 9.5. Partial regularity. 9.6. Notes and Comments -- ch. 10. Higher derivatives. 10.1. Hilbert regularity. 10.2. Constant coefficients. 10.3. Continuous coefficients. 10.4. L[symbol] estimates. 10.5. Minima of functionals. 10.6. Notes and comments. Calculus of variations. http://id.loc.gov/authorities/subjects/sh85018809 Calcul des variations. MATHEMATICS Calculus. bisacsh MATHEMATICS Mathematical Analysis. bisacsh Calculus of variations fast |
| subject_GND | http://id.loc.gov/authorities/subjects/sh85018809 |
| title | Direct methods in the calculus of variations / |
| title_alt | Calculus of variations |
| title_auth | Direct methods in the calculus of variations / |
| title_exact_search | Direct methods in the calculus of variations / |
| title_full | Direct methods in the calculus of variations / Enrico Giusti. |
| title_fullStr | Direct methods in the calculus of variations / Enrico Giusti. |
| title_full_unstemmed | Direct methods in the calculus of variations / Enrico Giusti. |
| title_short | Direct methods in the calculus of variations / |
| title_sort | direct methods in the calculus of variations |
| topic | Calculus of variations. http://id.loc.gov/authorities/subjects/sh85018809 Calcul des variations. MATHEMATICS Calculus. bisacsh MATHEMATICS Mathematical Analysis. bisacsh Calculus of variations fast |
| topic_facet | Calculus of variations. Calcul des variations. MATHEMATICS Calculus. MATHEMATICS Mathematical Analysis. Calculus of variations |
| url | https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=235657 |
| work_keys_str_mv | AT giustienrico directmethodsinthecalculusofvariations AT giustienrico calculusofvariations |