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2001 | Buch

The Monge—Ampère Equation

verfasst von: Cristian E. Gutiérrez

Verlag: Birkhäuser Boston

Buchreihe : Progress in Nonlinear Differential Equations and Their Applications

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In recent years, the study of the Monge-Ampere equation has received consider­ able attention and there have been many important advances. As a consequence there is nowadays much interest in this equation and its applications. This volume tries to reflect these advances in an essentially self-contained systematic exposi­ tion of the theory of weak: solutions, including recent regularity results by L. A. Caffarelli. The theory has a geometric flavor and uses some techniques from har­ monic analysis such us covering lemmas and set decompositions. An overview of the contents of the book is as follows. We shall be concerned with the Monge-Ampere equation, which for a smooth function u, is given by (0.0.1) There is a notion of generalized or weak solution to (0.0.1): for u convex in a domain n, one can define a measure Mu in n such that if u is smooth, then Mu 2 has density det D u. Therefore u is a generalized solution of (0.0.1) if M u = f.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Generalized Solutions to Monge-Ampère Equations
Abstract
Let Ω be an open subset of lib“ and u :Ω→ ℝ. Given x0 ∈ Ω, supporting hyperplane to the function u at the point (x0, u(x0))is an affine function ℓ(x = u(x0) + p · (xx0) such that u(x) ≥ ℓ(x) for all x ∈ Ω.
Cristian E. Gutiérrez
Chapter 2. Uniformly Elliptic Equations in Nondivergence Form
Abstract
In this chapter we consider linear operators of the form
$$ Lu = \sum\limits_{i,j = 1}^n {{a_{ij}}(x){D_{ij}}u(x)} $$
where the coefficient matrix A(x) = (aij (x)) is symmetric and uniformly elliptic, that is
$$\lambda {\left| \xi \right|^2} \leqslant \left\langle {A(x)\xi ,\xi } \right\rangle \leqslant \Lambda {\left| \xi \right|^2}$$
, for all ξ∈ℝ n and x∈Ω⊂ ℝn. We assume that the coefficients ai j are smooth functions, but the estimates we shall establish are independent of the regularity of the coefficients and depend only on the ellipticity constants λ, Λ and the dimension n.
Cristian E. Gutiérrez
Chapter 3. The Cross-sections of Monge—Ampère
Abstract
Let φ : ℝ n → ℝ be a convex function.
Cristian E. Gutiérrez
Chapter 4. Convex Solutions of detD2u = 1 in ℝ n
Abstract
We begin with an important and useful lemma due to Pogorelov.
Cristian E. Gutiérrez
5. Regularity Theory for the Monge-Ampère Equation
Abstract
Definition 5.1.1Let Ω be a convex subset of n . The point is an extremal point of Ω if x0 is not a convex combination of other points in \(\bar \Omega\).
Cristian E. Gutiérrez
6. W2,p Estimates for the Monge—Amperè Equation
Abstract
Our purpose in this chapter is to prove Caffarelli’s interiorLPestimates for second derivatives of solutions to the Monge—Ampère equation. That is, solutionsutoMu = fwithfpositive and continuous have second derivatives inLP,for 0 <p <∞, Theorem 6.4.2. The origin of these estimates goes back to Pogorelov [Pog71] who proved that convex solutions to detD 2 u = 1on a bounded convex domain Q withu =0 onasatisfy theL estimate
$$ {C_1}(\Omega ',\Omega )Id \leqslant {D^2}u(x) \leqslant {C_2}(\Omega ',\Omega )Id$$
(6.0.1)
, forxE Q’, where Q’ is a convex domain with closure contained in S2,Id isthe identity matrix, andCiare positive constants depending only on the domains. The estimates (6.0.1) have been proved in Chapter 4, and they follow as a consequence of Lemma 4.1.1; see (4.2.6).
Cristian E. Gutiérrez
Backmatter
Metadaten
Titel
The Monge—Ampère Equation
verfasst von
Cristian E. Gutiérrez
Copyright-Jahr
2001
Verlag
Birkhäuser Boston
Electronic ISBN
978-1-4612-0195-3
Print ISBN
978-1-4612-6656-3
DOI
https://doi.org/10.1007/978-1-4612-0195-3