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## Über dieses Buch

Investigations in modem nonlinear analysis rely on ideas, methods and prob­ lems from various fields of mathematics, mechanics, physics and other applied sciences. In the second half of the twentieth century many prominent, ex­ emplary problems in nonlinear analysis were subject to intensive study and examination. The united ideas and methods of differential geometry, topology, differential equations and functional analysis as well as other areas of research in mathematics were successfully applied towards the complete solution of com­ plex problems in nonlinear analysis. It is not possible to encompass in the scope of one book all concepts, ideas, methods and results related to nonlinear analysis. Therefore, we shall restrict ourselves in this monograph to nonlinear elliptic boundary value problems as well as global geometric problems. In order that we may examine these prob­ lems, we are provided with a fundamental vehicle: The theory of convex bodies and hypersurfaces. In this book we systematically present a series of centrally significant results obtained in the second half of the twentieth century up to the present time. Particular attention is given to profound interconnections between various divisions in nonlinear analysis. The theory of convex functions and bodies plays a crucial role because the ellipticity of differential equations is closely connected with the local and global convexity properties of their solutions. Therefore it is necessary to have a sufficiently large amount of material devoted to the theory of convex bodies and functions and their connections with partial differential equations.

## Inhaltsverzeichnis

### Chapter 1. Convex Bodies and Hypersurfaces

Abstract
The concept of a convex set can be introduced in any linear space L. A set K in L is called convex if the line segment ab is contained in K for any elements a, b ∈ K, i.e. $${x_{t}} = \left( {1 - t} \right)a + tb \in K$$ for any a, b ∈ K and any t ∈ [0,1].
Ilya J. Bakelman

### Chapter 2. Mixed Volumes. Minkowski Problem. Selected Global Problems in Geometric Partial Differential Equations

Abstract
Let Vn+1 be a (n + 1)-dimensional Euclidean vector space. We denote by En+1 the (n + 1)-dimensional Euclidean point space associated with Vn+1 (see §1). Then every vector aVn+1 generates a parallel translation
$${p_{a}}:{E^{{n + 1}}} \to {E^{{n + 1}}}$$
(7.1)
in the following way: for any point XEn+1
$${p_{a}}\left( X \right) = X',$$
, where X’ is the terminal point of the vector $$\overline {XX'} = a$$.
Ilya J. Bakelman

### Chapter 3. Generalized Solutions of N-Dimensional Monge-Ampere Equations

Abstract
First of all we introduce the concepts of the normal mapping of convex functions and the R-curvature of these functions. The R-curvature of convex functions is the extension of Monge-Ampere operators to the class of all general convex functions. We study in detail the properties of the normal mapping and R-curvature of convex functions and then investigate the solvability of the Dirichlet problem for weak and generalized elliptic solutions together with uniqueness and non-uniqueness theorems for these solutions.
Ilya J. Bakelman

### Chapter 4. Variational Problems and Generalized Elliptic Solutions of Monge-Ampere Equations

Abstract
There is a profound connection between n-dimensional variational problems and the Dirichlet problem for n-dimensional Monge-Ampere equations. The absolute minimum of these variational problems turns out to be a generalized solution of the corresponding Monge-Ampere equations. In this chapter we study explicitly the main variational problem connected with the Monge- Ampere equation
$$\det \left( {{u_{{ij}}}} \right) = f\left( {{x_{1}},{x_{2}}, \ldots ,{x_{n}}} \right)$$
(7.1)
and also consider generalizations of it.
Ilya J. Bakelman

### Chapter 5. Non-Compact Problems for Elliptic Solutions of Monge-Ampere Equations

Abstract
In this chapter we are concerned with elliptic generalized solutions of Monge-Ampere equations
$$\det \left( {{u_{{ij}}}} \right) = f\left( {x,u,Du} \right)$$
in the entire n-dimensional Euclidean space E n . Generalized elliptic solutions of equation (*) are convex or concave functions and their graphs are complete convex hypersurfaces in the space $${E^{{n + 1}}} = {E^{n}} \times R$$.These hypersurfaces project one-to-one on E n . Clearly it is sufficient to investigate only convex generalized solutions of equation (*) and confine oneself only to nonnegative functionsf (x,u,p)for all xE n , uR, pR n . As we know, any convex generalized solution u(x) of equation (*) satisfies this equation almost everywhere in any compact subset of E n and the set function
$$w\left( {l,u,e} \right) = meas {X_{u}}\left( e \right),$$
generated by u(x), is absolutely continuous on the family of Borel subsets of En.
Ilya J. Bakelman

### Chapter 6. Smooth Elliptic Solutions of Monge-Ampere Equations

Abstract
In § 8 of Chapter 2 we presented in detail the classical Minkowski Theorem on the problem of existence and uniqueness of a closed convex hypersurface with prescribed Gaussian curvature K(η) in (n + l)-dimensional Euclidean space E n+1 . Here K(η) is a positive continuous function on the unit hypersphere S n En+1, which is centered at the origin of En+1. The Minkowski problem is the problem of existence and uniqueness of a closed convex hypersurface F with Gaussian curvature K(η) at a point x with exterior unit normal η. Here we do not assume that F is a regular hypersurface. Therefore the Gaussian curvature of a hypersurface F at a point x ∈ F is defined as the limit of the ratio $$\frac{{w\left( G \right)}}{{w\left( G \right)}}$$ as domain G shrinks to the point x, where σ(G) is the area of G and ω(G) is the area of the spherical image of G. Both set functions σ(G) and ω(G) are defined in §§ 5, 8. This definition of Gaussian curvature does not assume the Cm-smoothness (m ≥ 2) of a convex hypersurface.
Ilya J. Bakelman

### Chapter 7. Geometric Concepts and Methods in Nonlinear Elliptic Euler-Lagrange Equations

Abstract
This chapter is concerned with global connections between the integrand F(x, u,p) of n-multiple integrals
$$I\left( u \right) = \int_{B} {F\left( {x,u\left( x \right),Du\left( x \right)} \right)} dx$$
and a priori estimates for solutions of the corresponding Euler-Lagrange equations, whose gradients satisfy some prescribed limitations. Such problems arise in the calculus of variations, differential geometry and continuum mechanics. Typical examples of these problems are presented in §§ 23, 24. Some of these estimates are called geometric maximum principles for variational problems.
Ilya J. Bakelman

### Chapter 8. The Geometric Maximum Principle for General Non-Divergent Quasilinear Elliptic Equations

Abstract
In this chapter we present the geometric maximum principles for solutions $$u\left( x \right) \in W_{2}^{n}\left( B \right) \cap C\left( {\overline B } \right)$$ of the elliptic Dirichlet problem
$$\sum\limits_{{i,j = 1}}^{n} {{a_{{ij}}}\left( {x,u,Du} \right){u_{{ij}}} = b\left( {x,u,Du} \right),}$$
(VIII.1)
$$u\left| {_{{\partial B}} = h\left( x \right),} \right.$$
(VIII.2)
where B is a bounded domain in the Euclidean space E n and ∂B is a closed continuous hypersurface in E n . We assume that $$h\left( x \right) \in C\left( {\partial B} \right)$$.
Ilya J. Bakelman

### Backmatter

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