Equation Of Monge-Ampere Type Research Articles

Light reflection is nonlinear optimization

In this paper, we show that the near field reflector problem is a nonlinear optimization problem. From the corresponding functional and constraint function, we derive the Monge–Ampere type equation for such a problem.

On self-similar blow-up in evolution equations of Monge-Ampere type

We use techniques from reaction-diffusion theory to study the blow-up and existence of solutions of the parabolic Monge–Ampere (M-A) equation with power source, with the following basic 2D model $$u_{t}=-|D^{2}u|+|u|^{p-1}u\quad \mathrm{in}\quad \mathbb{R}^{2}\times \mathbb{R}_{+},$$ where in two-dimensions $|D^{2}u|=u_{xx}u_{yy}-(v_{xy})^{2}$ and p > 1 is a fixed exponent. For a class of ‘dominated concave’ and compactly supported radial initial data $u_{0}(x)\geqslant0$ , the Cauchy problem is shown to be locally well posed and to exhibit finite time blow-up that is described by similarity solutions. For p ∈ (1, 2], similarity solutions, containing domains of concavity and convexity, are shown to be compactly supported and correspond to surfaces with flat sides that persist until the blow-up time. The case p > 2 leads to single-point blow-up. Numerical computations of blow-up solutions without radial symmetry are also presented. The parabolic analogy of the parabolic M-A equation in 3D for which $|D^{2}u|$ is a cubic operator is $$u_{t}=|D^{2}u|+|u|^{p-1}u\quad \mathrm{in}\quad \mathbb{R}^{3}\times \mathbb{R}_{+},$$ and is shown to admit a wider set of (oscillatory) self-similar blow-up patterns. Regional self-similarblow-up in a cubic radial model related to the fourth-order M-A equation $$u_{t}=-|D^{4}u|+u^{3}\quad \mathrm{in}\quad \mathbb{R}^{2}\times \mathbb{R}_{+},$$ where the cubic operator $|D^{4}u|$ is the catalecticant 3 × 3 determinant is also briefly discussed.

Designing Freeform Lenses for Intensity and Phase Control of Coherent Light with Help from Geometry and Mass Transport

We study here the problem of determining a system of two refractive interfaces transforming a plane wavefront of a given shape and radiation intensity into a coherent output plane wavefront with prescribed output position, shape and intensity. Such interfaces can be refracting surfaces of two different lenses or of one lens. In geometrical optics approximation, the analytic formulation of this problem in both cases requires construction of maps with controlled Jacobian. Though this Jacobian can be expressed as a second order partial differential equation of Monge-Ampere type for a scalar function defining one of the refracting surfaces, its analysis is not straightforward. In this paper we use a geometric approach for reformulating the problem in certain associated measures and defining weak solutions. Existence and uniqueness of weak solutions in Lipschitz classes for both cases are established by variational methods. Our results show, in particular, that two types of interfaces exist in each case for the same data: one of these types always consists of two interfaces, one of which is concave or convex and the second convex or concave, while the interfaces of the second type may be neither convex nor concave. The availability of a design with convex/concave lenses is particularly important for fabrication. The truly geometric nature of this problem permits its statement and investigation in \({\mathbb {R}^{N+1},\, N \geqq 1}\) .

Inductive schemes for the complete classification of affine hypersurfaces with parallel second fundamental form

We present in this article inductive schemes that allow to classify all those affine hypersurfaces with affine normal parallel second fundamental (cubic) form, for every possible dimensional case greater or equal than two. The solution proposed is a follow up to previous works on the same topic by this author and it uses, firstly, the reduction of the problem, eminently geometric, to the classification of a certain class of solutions to an equation of Monge-Ampere type. Then it is applied the so-called “method of algorithmic sequence of coordinate changes”, in order to achieve the latter.

On a modified parabolic complex Monge–Ampère equation with applications

We study a modified parabolic complex Monge–Ampere type equation on a complete non-compact Kahler manifold. We prove a short time existence result and obtain basic estimates. Applying these results, we prove that under certain assumptions on a given real and closed (1,1) form Ω and initial Kahler metric g0 on M, the modified Kahler–Ricci flow g′ = −Ric + Ω has a long time solution converging to a complete Kahler metric such that Ric = Ω, which extends the result in Cao (Invent Math 81:359–372, 1985) to non-compact manifolds. We will also obtain a long time existence result for the Kahler–Ricci flow which generalizes a result (Chau et al. in Can J Math, to appear).

Some Weighted Energy Classes of Plurisubharmonic Functions

In this paper we study relations between the weighted energy class \(\mathcal{E}_{\chi}\) introduced by S. Benelkourchi, V. Guedj and A. Zeriahi recently with Cegrell’s classes \(\mathcal{E}\) and \(\mathcal{N}\). Next we establish a generalized comparison principle for the operator Mχ. As an application, we prove a version of existence of solutions of Monge–Ampere type equations in the class \(\mathcal{E}_{\chi}(H,\Omega)\).

General solution of two-dimensional beam-shaping with two surfaces

Beam shaping is a technique, by which a known input irradiance is transformed into a desired output irradiance by changing the local propagation vector of the wave front. Unlike one-dimensional beam-shaping, which leads to a simple differential equation which can be integrated in a straight forward manner, the two-dimensional beam shaping problem leads to a Monge-Ampere type equation, which is difficult to solve. In this paper, we generalize the problem to refractive and reflective systems and use to shifted-base-function approach to obtain a general solution.

On a Monge–Ampère type equation in the Cegrell class Eχ

Let Omega be a bounded hyperconvex domain in C-n and let mu be a positive and finite measure which vanishes on all pluripolar subsets of Omega. We prove that for every continuous and strictly increasing function chi : (-infinity, 0) -> (-infinity, 0) there exists a negative plurisubharmonic function u which solves the Monge-Ampere type equation -chi(u)(dd(c)u)(n) = d mu. Under some additional assumption the solution u is uniquely determined.

A quadratically constrained minimization problem arising from PDE of Monge–Ampère type

This note develops theory and a solution technique for a quadratically constrained eigenvalue minimization problem. This class of problems arises in the numerical solution of fully-nonlinear boundary value problems of Monge–Ampère type. Though it is most important in the three dimensional case, the solution method is directly applicable to systems of arbitrary dimension. The focus here is on solving the minimization subproblem which is part of a method to numerically solve a Monge–Ampère type equation. These subproblems must be evaluated many times in this numerical solution technique and thus efficiency is of utmost importance. A novelty of this minimization algorithm is that it is finite, of complexity \(\mathcal{O}(n^3)\), with the exception of solving a very simple rational function of one variable. This function is essentially the same for any dimension. This result is quite surprising given the nature of the constrained minimization problem.

The Existence of Convex Body with Prescribed Curvature Measures

Curvature measure and surface area measure are the basic notions in the classical differential geometry. They play fundamental roles in the theory of convex bodies. They are closely related to the differential geometry and integral geometry of convex hypersurfaces. The Minkowski problem is the problem of prescribing n-th surface area measure on Sn. The Christoffel problem concerns the prescribing the 1-st surface area measure (e.g., see [1, 14, 17, 6, 19, 7, 3]). The general problem of prescribing surface area measures is called the Christoffel-Minkowski problem, we refer [12] for an updated account. The problem of prescribing 0-th curvature measure is called the Alexandrov problem, which is a counterpart to Minkowski problem. The problem is equivalent to solve a Monge-Ampere type equation on Sn. The existence and uniqueness were obtained by Alexandrov [2]. The regularity of the Alexandrov problem in elliptic case was proved by Pogorelov [18] for n = 2 and by Oliker [16] for higher dimension case. The general regularity results (degenerate case) of the problem were obtained in [9]. The general problem of prescribing (n− k)-th curvature measure for case k ≤ n is an interesting counterpart of the Christoffel-Minkowski problem. It has been discussed in literature (e.g., [20]). Nevertheless, very little is known except for the Alexandrov problem. In this paper, we are concerned with the existence of convex bodies with the prescribed (n− k)-th curvature measure for 1 ≤ k < n. We start with the definitions of curvature measures and surface area measures for convex bodies with smooth boundary. Let Ω be a bounded convex body in Rn+1 with C2 boundary M , the corresponding curvature measures and surface area measures of Ω can be defined according to some geometric quantities of M . Let κ = (κ1, · · · , κn) be the principal curvatures of M at point x, let Wk(x) = Sk(κ(x)) be the k-th Weingarten curvature of M at x (where Sk is the k-th elementary symmetric function). In particular, W1,W2

C 1 regularity of solutions of the Monge–Ampère equation for optimal transport in dimension two

We prove C1 regularity of c-convex weak Alexandrov solutions of a Monge–Ampere type equation in dimension two, assuming only a bound from above on the Monge–Ampere measure. The Monge–Ampere equation involved arises in the optimal transport problem. Our result holds true under a natural condition on the cost function, namely non-negative cost-sectional curvature, a condition introduced in Ma et al. (Arch Ration Mech Anal 177(2):151–183, 2005), that was shown in Loeper (Acta Math, to appear) to be necessary for C1 regularity. Such a condition holds in particular for the case “cost = distance squared” which leads to the usual Monge–Ampere equation det D2u = f. Our result is in some sense optimal, both for the assumptions on the density [thanks to the regularity counterexamples of Wang (Proc Am Math Soc 123(3):841–845, 1995)] and for the assumptions on the cost-function [thanks to the results of Loeper (Acta Math, to appear)].

Weak solutions of one inverse problem in geometric optics

We consider the problem of recovering a closed convex reflecting surface such that for a given point source of light (inside the convex body bounded by the surface) the reflected directions cover a unit sphere with prespecified in advance density. In analytic formulation, the problem leads to an equation of Monge-Ampere type on a unit sphere. We formulate the problem in terms of certain associated measures and establish the existence of weak solutions. Bibliography: 11 titles.

Bubbling AdS and droplet descriptions of BPS geometries in IIB supergravity

This paper focuses on supergravity duals of BPS states in = 4 super Yang-Mills. In order to describe these duals, we begin with a sequence of breathing mode reductions of IIB supergravity: first on S3, then S3 × S1, and finally on S3 × S1 × CP1. We then follow with a complete supersymmetry analysis, yielding 1/8, 1/4 and 1/2 BPS configurations, respectively (where in the last step we take the Hopf fibration of S3). The 1/8 BPS geometries, which have an S3 isometry and are time-fibered over a six-dimensional base, are determined by solving a non-linear equation for the Kahler metric on the base. Similarly, the 1/4 BPS configurations have an S3 × S1 isometry and a four-dimensional base, whose Kahler metric obeys another non-linear, Monge-Ampere type equation. Despite the non-linearity of the problem, we develop a universal bubbling AdS description of these geometries by focusing on the boundary conditions which ensure their regularity. In the 1/8 BPS case, we find that the S3 cycle shrinks to zero size on a five-dimensional locus inside the six-dimensional base. Enforcing regularity of the full solution requires that the interior of a smooth, generally disconnected five-dimensional surface be removed from the base. The AdS5 × S5 ground state corresponds to excising the interior of an S5, while the 1/8 BPS excitations correspond to deformations (including topology change) of the S5 and/or the excision of additional droplets from the base. In the case of 1/4 BPS configurations, by enforcing regularity conditions, we identify three-dimensional surfaces inside the four-dimensional base which separate the regions where the S3 shrinks to zero size from those where the S1 shrinks. We discuss a large class of examples to show the emergence of a universal bubbling AdS picture for all 1/2, 1/4 and 1/8 BPS geometries.

A Bernstein property of solutions to a class of prescribed affine mean curvature equations

Let $$x: M \rightarrow A^{n+1}$$ be a locally strongly convex hypersurface, given as the graph of a locally strongly convex function x n+1 = z(x 1, ..., x n ). In this paper we prove a Bernstein property for hypersurfaces which are complete with respect to the metric $$G^{\sharp} = \sum \left( \frac{\partial^{2}z}{\partial x_{i} \partial x_{j}} \right) dx_{i} dx_{j}$$ and which satisfy a certain Monge–Ampere type equation. This generalises in some sense the earlier result of Li and Jia for affine maximal hypersurfaces of dimension n = 2 and n = 3 (Li, A.-M., Jia, F.: A Bernstein property of affine maximal hypersurfaces. Ann. Glob. Anal. Geom. 23, 359–372 (2003)), related results (Li, A.-M., Jia, F.: Locally strongly convex hypersurfaces with constant affine mean curvature. Diff. Geom. Appl. 22(2), 199–214 (2005)) and results for n = 2 of Trudinger and Wang (Trudinger, N.S., Wang, X.-J.: Bernstein-Jorgens theorem for a fourth order partial differential equation. J. Partial Diff. Equ. 15(2), 78–88 (2002)).

The equation of complex Monge-Ampère type and stability of solutions

We prove that in a family of plurisubharmonic functions with Monge-Ampere measures bounded from above by such a measure of one function weak convergence is equivalent to convergence in capacity. We also show a very general statement on the existence of solutions of the complex Monge-Ampere type equation.

Monge-Ampère foliations with singularities at the boundary of strongly convex domains

Let D ⊂ C N be a bounded strongly convex domain with smooth boundary. We consider a Monge-Ampere type equation in D with a simple pole at the boundary. Using the Lempert foliation of D in extremal discs, we construct a solution u whose level sets are bound- aries of horospheres. Among other things, we show that the biholomorphisms between strongly convex domains are exactly those maps which preserves our solution.

Regularity of Potential Functions of the Optimal Transportation Problem

The potential function of the optimal transportation problem satisfies a partial differential equation of Monge-Ampere type. In this paper we introduce the notion of a generalized solution, and prove the existence and uniqueness of generalized solutions of the problem. We also prove the solution is smooth under certain structural conditions on the cost function.

Comparison results for Monge-Amp�re type equations with lower order terms

In this paper we deal with Monge-Ampere type equations in two dimensions and, using the symmetrization with respect to the perimeter, we prove some comparison results for solutions of such equations involving the solutions of conveniently symmetrized problems.

Approximation results by difference schemes of fracture energies: the vectorial case

In this paper we deal with Monge-Ampere type equations in two dimensions and, using the symmetrization with respect to the perimeter, we prove some comparison results for solutions of such equations involving the solutions of conveniently symmetrized problems.

On affine hypersurfaces with parallel second fundamental form

We investigate the classification problem of hypersurfaces with affine normal parallel second fundamental (cubic) form. A new method of approaching the solution to this problem is here presented; it consists in showing and using the equivalence of the mentioned problem with the classification of a certain class of solutions to the equation of Monge-Ampere type $\det(\pa_{ij}f)=\pm1$.