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Monge-Ampere Equation 2001 ed. [Kietas viršelis]

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Monge-Ampere Equation 2001 ed.Didesnis paveikslėlis
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The Monge-Ampere equation has attracted considerable interest in recent years because of its important role in several areas of applied mathematics. Monge-Ampere type equations have applications in the areas of differential geometry, the calculus of variations, and several optimization problems, such as the Monge-Kantorovitch mass transfer problem. This book stresses the geometric aspects of this beautiful theory, using techniques from harmonic analysis - covering lemmas and set decompositions.

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Preface ix
Notation xi
Generalized Solutions to Monge-Ampere Equations
1(30)
The normal mapping
1(5)
Properties of the normal mapping
2(4)
Generalized solutions
6(2)
Viscosity solutions
8(2)
Maximum principles
10(7)
Aleksandrov's maximum principle
11(1)
Aleksandrov-Bakelman-Pucci's maximum principle
12(4)
Comparison principle
16(1)
The Dirichlet problem
17(2)
The nonhomogeneous Dirichlet problem
19(5)
Return to viscosity solutions
24(1)
Ellipsoids of minimum volume
25(4)
Notes
29(2)
Uniformly Elliptic Equations in Nondivergence Form
31(14)
Critical density estimates
31(6)
Estimate of the distribution function of solutions
37(3)
Harnack's inequality
40(3)
Notes
43(2)
The Cross-sections of Monge-Ampere
45(18)
Introduction
45(2)
Preliminary results
47(3)
Properties of the sections
50(12)
The Monge-Ampere measures satisfying (3.1.1)
50(5)
The engulfing property of the sections
55(2)
The size of normalized sections
57(5)
Notes
62(1)
Convex Solutions of det D2u = 1 in Rn
63(12)
Pogorelov's Lemma
63(4)
Interior Holder estimates of D2u
67(3)
Cα estimates of D2u
70(4)
Notes
74(1)
Regularity Theory for the Monge-Ampere Equation
75(20)
External points
75(2)
A result on extremal points of zeroes of solutions to Monge-Ampere
77(2)
A strict convexity result
79(5)
C1,α regularity
84(8)
Examples
92(1)
Notes
93(2)
W2,p Estimates for the Monge-Ampere Equation
95(28)
Approximation Theorem
95(4)
Tangent paraboloids
99(2)
Density estimates and power decay
101(7)
Lp estimates of second derivatives
108(4)
Proof of the Covering Theorem 6.3.3
112(7)
Regularity of the convex envelope
119(2)
Notes
121(2)
Bibliography 123(4)
Index 127