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The Monge-Ampère equation. (English) Zbl 0989.35052

Progress in Nonlinear Differential Equations and their Applications. 44. Boston, MA: Birkhäuser. xi, 127 p. (2001).
In recent years the Monge-Ampère equation \[ \det D^2u = f \tag{1} \] has received a great deal of attention. Originally this equation arose in differential geometry, in connection with problems involving hypersurfaces of prescribed Gauss curvature, such as the Minkowski problem. More general Monge-Ampère equations also arise in geometry, for example, in the Weyl embedding problem. More recently a connection with mass transportation problems was discovered, and this has led to renewed interest in Monge-Ampère equations in the calculus of variations and optimization theory, as well as in applications such as meteorology, fluid dynamics, and mathematical physics.
In this book the author presents an essentially self-contained exposition of the theory of generalized solutions of (1), starting with some basic facts about convex functions, and ending with the deep interior regularity theory developed by L. A. Caffarelli [Ann. Math., II. Ser. 131, No. 1, 129-134 (1990; Zbl 0704.35045), ibid., 135-150 (1990; Zbl 0704.35044)]. Aspects of the theory dealing with the existence of globally smooth solutions are not discussed. Along the way some fundamental results for solutions of linear elliptic equations are also proved.
Chapter 1 contains a very clear account of the theory of Aleksandrov generalized solutions of (1). This provides a way of interpreting \(\det D^2u\) as a Borel measure on \(\Omega\) if \(u\) is a convex function defined on a convex domain \(\Omega\subset{\mathbb R}^n\). One can then study the solvability of the Dirichlet problem for (1) in the class of convex functions. A central result of Chapter 1 is Theorem 1.6.2, which states that for any bounded strictly convex domain \(\Omega\subset{\mathbb R}^n\), any Borel measure \(\mu\) on \(\Omega\) with \(\mu(\Omega)<\infty\), and any \(g\in C(\partial\Omega)\), the equation \(\det D^2u=\mu\) has a unique convex generalized solution \(u\) with \(u=g\) on \(\partial\Omega\).
Chapter 1 also deals with the Aleksandrov-Bakelman-Pucci maximum principle. One variant of this states that if \(u\in C^2(\Omega)\cap C(\overline\Omega)\), then \[ \max_{\Omega} u \leq \max_{\partial\Omega} u + \omega_n^{-1/n}\text{diam}(\Omega) \left( \int_{{\mathcal C}^*(u)} |\det D^2 u(x)|dx \right)^{1/n}, \] where \(\omega_n\) is the Lebesgue measure of the unit ball in \({\mathbb R}^n\) and \[ {\mathcal C}^*(u) = \{x\in\Omega : u(y)\leq u(x) + Du(x)\cdot(y-x) \text{ for all } y\in\Omega \} \] is the upper contact set of \(u\). This estimate is essentially an estimate for convex solutions of (1). Its variants and extensions to \(u\in C(\overline\Omega)\) are central to the regularity theory developed later in the book, and are also important in the theory of linear elliptic equations.
The chapter concludes with a discussion of ellipsoids of least volume containing a given bounded convex set \(\Omega\subset{\mathbb R}^n\) with nonempty interior. The existence of such an ellipsoid is proved, and furthermore, it is shown that if \(E\) is an ellipsoid of minimum volume containing \(\Omega\) and centred at the centre of mass of \(\Omega\), then \(\alpha_nE \subset \Omega \subset E\) where \(\alpha_n=n^{-3/2}\) and \(\alpha E\) denotes the \(\alpha\)-dilation of \(E\) with respect to its centre. If \(T\) is an affine transformation such that \(T(E)\) is the unit ball in \({\mathbb R}^n\), then the set \(T(\Omega)\) is said to be normalized by \(T\). Such normalizations arise naturally when studying the Monge-Ampère equation, because \(\det D^2u\) is invariant with respect to affine transformations of determinant one. This invariance is strongly exploited throughout the book.
Chapter 2 is concerned with the derivation of the fundamental Harnack inequality of N. V. Krylov and M. V. Safonov [Izv. Akad. Nauk SSSR, Ser. Mat. 44, 161-175 (1980; Zbl 0439.35023)] for solutions of linear elliptic equations of the form \[ \sum_{i,j=1}^n a^{ij}D_{ij}u = f \tag{2} \] where \(a^{ij}\) is symmetric, measurable and satisfies \[ \lambda|\xi|^2 \leq \sum_{i,j=1}^n a^{ij}\xi_i\xi_j \leq \Lambda|\xi|^2 \quad\text{for all}\quad \xi\in{\mathbb R}^n \tag{3} \] for some positive constants \(\lambda,\Lambda\).
In Chapter 3 sections of convex functions are studied. A section of a convex function \(\phi\) defined on \({\mathbb R}^n\) is a set of the form \[ S_{\phi}(x_0,p,t) = \{x\in{\mathbb R}^n:\phi(x)<\ell(x)+t \} \] where \(t>0\) and \(\ell(x)=\phi(x_0)+p\cdot(x-x_0)\) is a supporting hyperplane to the graph of \(\phi\) at \((x_0,\phi(x_0))\). It is assumed that \(\phi\) is such that \(S_{\phi}(x_0,p,t)\) is a bounded set for any \(x_0\in {\mathbb R}^n\) and any \(t>0\). Several important results are proved. First, a geometric characterization of Monge-Ampère measures \(\mu=M\phi\) that are doubling with respect to centre of mass is proved. This means that there are constants \(C>0\) and \(\alpha\in(0,1)\) such that for all sections \(S_{\phi}(x,p,t)\) we have \[ \mu(S_{\phi}(x,p,t)) \leq C\mu(\alpha S_{\phi}(x,p,t)), \] where \(\alpha S_{\phi}(x,p,t)\) denotes the \(\alpha\)-dilation of \(S_{\phi}(x,p,t)\) with respect to its centre of mass. It is shown that this is equivalent to the existence of \(\tau,\lambda\in(0,1)\) such that for all \(x_0\in{\mathbb R}^n\) and \(t>0\) we have \[ S_{\phi}(x_0,p,\tau t) \subset \lambda S_{\phi}(x_0,p,t). \] Furthermore, it is shown that if \(M\phi\) is doubling with respect to centre of mass, then sections of \(\phi\) behave like Euclidean balls in certain respects. These technical facts are of fundamental importance in the development of the subsequent regularity theory in particular, in the proof of a key covering lemma of Besicovitch type in Chapter 6.
Although not discussed in the book, the results on sections are also used in the study of the linearized Monge-Ampère equation, for example, in proving an affine invariant analogue of the Krylov-Safonov Harnack inequality for solutions of such equations. In this theory, developed by L. A. Caffarelli and the author [Trans. Am. Math. Soc. 384, No. 3, 1075-1092 (1996; Zbl 0858.35034)], the coefficient matrix \([a^{ij}]\) in (2) is the cofactor matrix of the Hessian of a convex function \(\phi\), and the uniform ellipticity condition (3) is replaced by \[ 0 < \lambda \leq \det D^2\phi \leq \Lambda < \infty. \tag{4} \] Further applications of the theory of sections arise in harmonic analysis.
In Chapter 4 the author gives a proof of a classical result of Jörgens (\(n=2\)), Calabi (\(n\leq 5\)), Pogorelov and Cheng and Yau (\(n\geq 2\)), that an entire convex solution of \(\det D^2u=1\) in \({\mathbb R}^n\) must be a paraboloid. Here a proof is given following an argument of Caffarelli, making use of a priori estimates and normalization techniques.
The last two chapters are concerned with the regularity of generalized solutions of (1). Chapter 5 contains a proof of the fact (due to Caffarelli) that if \(u\geq 0\) is a convex function on a bounded convex open set \(\Omega\subset{\mathbb R}^n\) such that \[ 0 < \lambda \leq \det D^2u \leq \Lambda < \infty \tag{5} \] in the generalized sense, and \(\Gamma=\{x\in\Omega: u(x)=0 \}\) is nonempty, then \(\Gamma\) has no extreme points in the interior of \(\Omega\). It is then shown that generalized solutions \(u\) of (5) with \(u=0\) on \(\partial\Omega\) are strictly convex and belong to \(C^{1,\alpha}(\Omega)\) for some \(\alpha>0\) depending only on \(\lambda,\Lambda\) and \(n\). An important point should be made here. Generalized solutions of (1) need not be even \(C^1\) in the interior of their domain of definition if \(n\geq 3\), even if \(f\) is smooth and positive. One needs some control of the boundary behaviour of \(u\) to prove interior regularity. An important technique for proving regularity of solutions of Monge-Ampère equations in many situations is to show that the solution is strictly convex; one can then easily localize to the case of affine Dirichlet data.
Chapter 6 contains an exposition of the interior regularity theory of Caffarelli for generalized solutions \(u\) of \[ \det D^2u = f(x) \quad\text{in}\quad\Omega, \qquad u = 0 \quad\text{on}\quad\partial\Omega. \] The basic result is that if \(f\) is continuous on \(\Omega\) and bounded between two positive constants, then \(u\) belongs to \(W^{2,p}_{\text{loc}}(\Omega)\) for all \(p<\infty\). The proof of this is very technical and uses much of the machinery developed in the earlier chapters.
In conclusion, the author has given a careful exposition of some of the main aspects of the theory of the Monge-Ampère equation. The book will be useful both to those seeking an introduction to the subject, and to researchers in the field of nonlinear partial differential equations.

MSC:

35J60 Nonlinear elliptic equations
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35J65 Nonlinear boundary value problems for linear elliptic equations
53A15 Affine differential geometry
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
35D05 Existence of generalized solutions of PDE (MSC2000)
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