The Monge-Ampère equation / Cristian E. Gutiérrez.
Now in its second edition, this monograph explores the Monge-Ampère equation and the latest advances in its study and applications. It provides an essentially self-contained systematic exposition of the theory of weak solutions, including regularity results by L.A. Caffarelli. The geometric aspects...
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| Online Access: |
Full Text (via Springer) |
|---|---|
| Main Author: | |
| Format: | eBook |
| Language: | English |
| Published: |
Switzerland :
Birkhäuser,
[2016]
|
| Edition: | Second edition. |
| Series: | Progress in nonlinear differential equations and their applications ;
v. 89. |
| Subjects: |
MARC
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| 100 | 1 | |a Gutiérrez, Cristian E., |d 1950- |e author. |0 http://id.loc.gov/authorities/names/n2001006714 |1 http://isni.org/isni/0000000114557836. | |
| 245 | 1 | 4 | |a The Monge-Ampère equation / |c Cristian E. Gutiérrez. |
| 250 | |a Second edition. | ||
| 264 | 1 | |a Switzerland : |b Birkhäuser, |c [2016] | |
| 264 | 4 | |c ©2016. | |
| 300 | |a 1 online resource (xiv, 216 pages) : |b illustrations (some color) | ||
| 336 | |a text |b txt |2 rdacontent. | ||
| 337 | |a computer |b c |2 rdamedia. | ||
| 338 | |a online resource |b cr |2 rdacarrier. | ||
| 490 | 1 | |a Progress in nonlinear differential equations and their applications, |x 2374-0280 ; |v volume 89. | |
| 504 | |a Includes bibliographical references (pages 211-214) and index. | ||
| 505 | 0 | 0 | |g 1. |t Generalized Solutions to Monge-Ampère Equations. |g 1.1.The |t Normal Mapping. |g 1.1.1. |t Properties of the Normal Mapping. |g 1.2. |t Generalized Solutions. |g 1.3. |t Viscosity Solutions. |g 1.4. |t Maximum Principles. |g 1.4.1. |t Aleksandrov's Maximum Principle. |g 1.4.2. |t Aleksandrov-Bakelman-Pucci's Maximum Principle. |g 1.4.3. |t Comparison Principle. |g 1.5.The |t Dirichlet Problem. |g 1.6.The |t Nonhomogeneous Dirichlet Problem. |g 1.7. |t Return to Viscosity Solutions. |g 1.8. |t Ellipsoids of Minimum Volume. |g 1.9. |t Exercises. |g 1.10. |t Notes -- |
| 505 | 8 | 0 | |g 2. |t Uniformly Elliptic Equations in Nondivergence Form. |g 2.1. |t Critical Density Estimates. |g 2.2. |t Estimate of the Distribution Function of Solutions. |g 2.3. |t Harnack's Inequality. |g 2.4. |t Notes -- |
| 505 | 8 | 0 | |g 3.The |t Cross-Sections of Monge-Ampère. |g 3.1.ntroduction. |g 3.2. |t Preliminary Results. |g 3.3. |t Properties of the Sections. |g 3.3.1. |t The Monge-Ampère Measures Satisfying (3.1.1). |g 3.3.2. |t The Engulfing Property of the Sections. |g 3.3.3. |t The Size of Normalized Sections. |g 3.4. |t Notes -- |
| 505 | 8 | 0 | |6 880-01 |g 6. |t W²,p Estimates for the Monge-Ampère Equation. |g 6.1. |t Approximation Theorem. |g 6.2. |t Tangent Paraboloids. |g 6.3. |t Density Estimates and Power Decay. |g 6.4. |t Lp Estimates of Second Derivatives. |g 6.5. |t Proof of the Covering Theorem 6.3.3. |g 6.6. |t Regularity of the Convex Envelope. |g 6.7. |t Notes -- |
| 505 | 8 | 0 | |g 7.The |t Linearized Monge-Ampère Equation. |g 7.1.Introduction. |g 7.2. |t Normalized Solutions. |g 7.3. |t Critical Density. |g 7.4. |t Double Section Property. |g 7.4.1. |t A[infinity] Condition on Sections. |g 7.4.2. |t Behavior of nonnegative Solutions in Expanded Sections. |g 7.5.A |t Calderón-Zygmund Type Decomposition for Sections. |g 7.6. |t Power Decay. |g 7.7. |t Interior Harnack's Inequality. |g 7.8. |t Notes -- |
| 520 | |a Now in its second edition, this monograph explores the Monge-Ampère equation and the latest advances in its study and applications. It provides an essentially self-contained systematic exposition of the theory of weak solutions, including regularity results by L.A. Caffarelli. The geometric aspects of this theory are stressed using techniques from harmonic analysis, such as covering lemmas and set decompositions. An effort is made to present complete proofs of all theorems, and examples and exercises are offered to further illustrate important concepts. Some of the topics considered include generalized solutions, non-divergence equations, cross sections, and convex solutions. New to this edition is a chapter on the linearized Monge-Ampère equation and a chapter on interior Hölder estimates for second derivatives. Bibliographic notes, updated and expanded from the first edition, are included at the end of every chapter for further reading on Monge-Ampère-type equations and their diverse applications in the areas of differential geometry, the calculus of variations, optimization problems, optimal mass transport, and geometric optics. Both researchers and graduate students working on nonlinear differential equations and their applications will find this to be a useful and concise resource. | ||
| 588 | 0 | |a Online resource; title from PDF title page (SpringerLink, viewed October 26, 2016) | |
| 650 | 0 | |a Monge-Ampère equations. |0 http://id.loc.gov/authorities/subjects/sh85086814. | |
| 650 | 7 | |a Monge-Ampère equations. |2 fast |0 (OCoLC)fst01025360. | |
| 776 | 0 | 8 | |i Print version: |a Gutiérrez, Cristian E., 1950- |t Monge-Ampère equation. |b Second edition. |d [Switzerland] : Birkhäuser, [2016] |z 9783319433721 |w (DLC) 2016950029. |
| 830 | 0 | |a Progress in nonlinear differential equations and their applications ; |v v. 89. |0 http://id.loc.gov/authorities/names/n88540928. | |
| 856 | 4 | 0 | |u https://colorado.idm.oclc.org/login?url=http://link.springer.com/10.1007/978-3-319-43374-5 |z Full Text (via Springer) |
| 880 | 8 | 0 | |6 505-00/(S |g 4. |t Convex Solutions of detD²u D 1 in Rn. |g 4.1. |t Pogorelov's Lemma. |g 4.2. |t Interior Hölder Estimates of D²2u. |g 4.3. |t Cα Estimates of D²2u. |g 4.4. |t Notes -- |
| 880 | 8 | 0 | |6 505-01/(S |g 5. |t Regularity Theory for the Monge-Ampère Equation. |g 5.1. |t Extremal Points. |g 5.2. |t A result on extremal points of zeroes of solutions to Monge-Ampère. |g 5.3. |t A Strict Convexity Result. |g 5.4. |t C¹,α Regularity. |g 5.4.1. |t C¹,α Estimates. |g 5.5. |t Examples. |g 5.5.1. |t A Generalization of Formula (5.5.1). |g 5.6. |t Notes -- |
| 880 | 8 | 0 | |6 505-00/(S |g 8. |t Interior Hölder Estimates for Second Derivatives. |g 8.1. |t Introduction. |g 8.2. |t Interior C2,α Estimates. |g 8.3. |t Notes. |
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