The Monge-Ampère Equation / by Cristian E. Gutiérrez.
The classical Monge-Ampère equation has been the center of considerable interest in recent years because of its important role in several areas of applied mathematics. In reflecting these developments, this works stresses the geometric aspects of this beautiful theory, using some techniques from har...
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Full Text (via Springer) |
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| Format: | eBook |
| Language: | English |
| Published: |
Boston, MA :
Birkhäuser Boston : Imprint : Birkhäuser,
2001.
|
| Series: | Progress in nonlinear differential equations and their applications ;
44. |
| Subjects: |
Table of Contents:
- 1 Generalized Solutions to Monge-Ampere Equations
- 1.1 The normal mapping
- 1.2 Generalized solutions
- 1.3 Viscosity solutions
- 1.4 Maximum principles
- 1.5 The Dirichlet problem
- 1.6 The nonhomogeneous Dirichlet problem
- 1.7 Return to viscosity solutions
- 1.8 Ellipsoids of minimum volume
- 1.9 Notes
- 2 Uniformly Elliptic Equations in Nondivergence Form
- 2.1 Critical density estimates
- 2.2 Estimate of the distribution function of solutions
- 2.3 Harnack's inequality
- 2.4 Notes
- 3 The Cross-sections of Monge-Ampere
- 3.1 Introduction
- 3.2 Preliminary results
- 3.3 Properties of the sections
- 3.4 Notes
- 4 Convex Solutions of det D2u = 1 in?n
- 4.1 Pogorelov's Lemma
- 4.2 Interior Hölder estimates of D2u
- 4.3 C?estimates of D2u
- 4.4 Notes
- 5 Regularity Theory for the Monge-Ampère Equation
- 5.1 Extremal points
- 5.2 A result on extremal points of zeroes of solutions to Monge-Ampère
- 5.3 A strict convexity result
- 5.4 C1,?regularity
- 5.5 Examples
- 5.6 Notes
- 6 W2pEstimates for the Monge-Ampere Equation
- 6.1 Approximation Theorem
- 6.2 Tangent paraboloids
- 6.3 Density estimates and power decay
- 6.4 LP estimates of second derivatives
- 6.5 Proof of the Covering Theorem 6.3.3
- 6.6 Regularity of the convex envelope
- 6.7 Notes.