Cristian E. Gutiérrez

The Monge-Ampère equation

1 Generalized Solutions to Monge-Ampere Equations.- 1.1 The normal mapping.- 1.1.1 Properties of the normal mapping.- 1.2 Generalized solutions.- 1.3 Viscosity solutions.- 1.4 Maximum principles.- 1.4.1 Aleksandrovs maximum principle.- 1.4.2 Aleksandrov-Bakelman-Puccis maximum principle.- 1.4.3 Comparison principle.- 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 Harnacks inequality.- 2.4 Notes.- 3 The Cross-sections of Monge-Ampere.- 3.1 Introduction.- 3.2 Preliminary results.- 3.3 Properties of the sections.- 3.3.1 The Monge-Ampere measures satisfying (3.1.1).- 3.3.2 The engulfing property of the sections.- 3.3.3 The size of normalized sections.- 3.4 Notes.- 4 Convex Solutions of det D2u = 1 in ?n.- 4.1 Pogorelovs Lemma.- 4.2 Interior Holder estimates of D2u.- 4.3 C?estimates of D2u.- 4.4 Notes.- 5 Regularity Theory for the Monge-Ampere Equation.- 5.1 Extremal points.- 5.2 A result on extremal points of zeroes of solutions to Monge-Ampere.- 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.

Weighted sobolev-poincaré inequalities for grushin type operators

(1994). Weighted sobolev-poincare inequalities for grushin type operators. Communications in Partial Differential Equations: Vol. 19, No. 3-4, pp. 523-604.

Real analysis related to the Monge-Ampère equation

In this paper we consider a family of convex sets in Rn, F = {S(x, t)}, x ∈ Rn, t > 0, satisfying certain axioms of affine invariance, and a Borel measure μ satisfying a doubling condition with respect to the family F . The axioms are modelled on the properties of the solutions of the real Monge-Ampère equation. The purpose of the paper is to show a variant of the Calderón-Zygmund decomposition in terms of the members of F . This is achieved by showing first a Besicovitch-type covering lemma for the family F and then using the doubling property of the measure μ. The decomposition is motivated by the study of the properties of the linearized Monge-Ampère equation. We show certain applications to maximal functions, and we prove a John and Nirenberg-type inequality for functions with bounded mean oscillation with respect to F .

Maximum and Comparison Principles for Convex Functions on the Heisenberg Group

Abstract We prove estimates, similar in form to the classical Aleksandrov estimates, for a Monge–Ampère type operator on the Heisenberg group. A notion of normal mapping does not seem to be available in this context and the method of proof uses integration by parts and oscillation estimates that lead to the construction of an analogue of Monge–Ampère measures for convex functions in the Heisenberg group.

The near field refractor

Abstract We present an abstract method in the setting of compact metric spaces which is applied to solve a number of problems in geometric optics. In particular, we solve the one source near field refraction problem. That is, we construct surfaces separating two homogeneous media with different refractive indices that refract radiation emanating from the origin into a target domain contained in an n − 1 dimensional hypersurface. The input and output energy are prescribed. This implies the existence of lenses focusing radiation in a prescribed manner.

The Parallel Refractor

Given two homogenous and isotropic media I and II with different refractive indices n I and n II , respectively, we have a source Ω surrounded by media I and a target screen Σ surrounded by media II. We prove existence of interface surfaces between the media that refract collimated radiation emanating from Ω into Σ with prescribed input and output intensities.

Singular integrals related to the Monge-Ampère equation

The purpose of this note is to describe some results of real analysis related with the Monge-Ampere equation that are proved in [1] and to show its application to the boundedness of certain singular integrals.

Harnack Inequality for a Degenerate Elliptic Equation

We consider degenerate elliptic equations of the form where X i, j are defined with the Heisenberg vector fields, and the matrix coefficient is uniformly elliptic. We obtain an invariant Harnacks inequality on metric balls for nonnegative solutions under the additional assumption that the ratio between the maximum and minimum eigenvalues of the coefficient matrix is sufficiently close to one. In the paper we prove critical density and double ball estimates. Once this is established, Harnack follows directly from the results from [4].

Refraction Problems in Geometric Optics

We present a description of the far field and the near field problems for refraction when the source of energy is located at one point. The far field problem is solved using mass transportation and also a variant of the Minkowski method. Maxwell equations are developed and the boundary conditions studied to obtain Fresnel formulas. These are used to present a model for refraction that takes into consideration the energy used in internal reflection.

Aspherical lens design

Given a three-dimensional surface G, not necessarily rotationally symmetric, and away from a point source, we design a surface F such that the lens sandwiched between the two surfaces refracts radiation into a given direction or into a given point. The surface F satisfies a system of first-order partial differential equations that can be solved in terms of G and the refractive indices of the media involved.