Monge-Ampere Research Articles

Legendre Transformation in the Born–Infeld Model, Monge–Ampere Equation and Exact Solutions

Legendre Transformation in the Born–Infeld Model, Monge–Ampere Equation and Exact Solutions

Monge-Ampère equations on compact Hessian manifolds

We consider degenerate Monge-Ampere equations on compact Hessian manifolds. We establish compactness properties of the set of normalized quasi-convex functions and show local and global comparison principles for twisted Monge-Ampere operators. We then use the Perron method to solve Monge-Ampere equations whose RHS involves an arbitrary probability measure, generalizing works of Cheng-Yau, Delanoe, Caffarelli-Viaclovsky and Hultgren-Onnheim. The intrinsic approach we develop should be useful in deriving similar results on mildly singular Hessian varieties, in line with the Strominger-Yau-Zaslow conjecture.

Uniqueness of Nontrivial Solutions for Degenerate Monge–Ampere Equations

Uniqueness of Nontrivial Solutions for Degenerate Monge–Ampere Equations

Moser-Trudinger inequalities and complex Monge-Ampère equation

Our aim is to give a version of the Moser-Trudinger inequality in the setting of complex geometry. As a very particular case, our result already gives a new Moser-Trudinger inequality for functions in the Sobolev space $W^{1,2}$ of a domain in $R^2$. We also deduce a new necessary condition for the complex Monge-Ampere equation for a given measure on a compact Kahler manifold to admit a Holder continuous solution.

A fast dynamic smooth adaptive meshing scheme with applications to compressible flow

We develop a fast-running smooth adaptive meshing (SAM) algorithm for dynamic curvilinear mesh generation, which is based on a fast solution strategy of the time-dependent Monge-Ampère (MA) equation, det⁡∇ψ(x,t)=G∘ψ(x,t). The novelty of our approach is a new so-called perturbation formulation of MA, which constructs the solution map ψ via composition of a sequence of near-identity deformations of a reference mesh. Then, we formulate a new version of the deformation method [21] that results in a simple, fast, and high-order accurate numerical scheme and a dynamic SAM algorithm that is of optimal complexity when applied to time-dependent mesh generation for solutions to hyperbolic systems such as the Euler equations of gas dynamics. We perform a series of challenging 2D and 3D mesh generation experiments for grids with large deformations, and demonstrate that SAM is able to produce smooth meshes comparable to state-of-the-art solvers [22,18], while running approximately 200 times faster. The SAM algorithm is then coupled to a simple Arbitrary Lagrangian Eulerian (ALE) scheme for 2D gas dynamics. Specifically, we implement the C-method [64,65] and develop a new ALE interface tracking algorithm for contact discontinuities. We perform numerical experiments for both the Noh implosion problem as well as a classical Rayleigh-Taylor instability problem. Results confirm that low-resolution simulations using our SAM-ALE algorithm compare favorably with high-resolution uniform mesh runs.

Three‐dimensional single‐shell dielectric lens design using complex coordinates and local axis‐displaced confocal quadrics

AbstractThis study proposes an analytical formulation in complex coordinates to synthesise three‐dimensional single‐shell dielectric lens surfaces. An exact formulation based on geometrical optics is developed, and the synthesis problem is modelled as a non‐linear second‐order partial differential equation of the Monge–Ampère (MA) type. An alternative numerical scheme based on axis‐displaced confocal quadrics is used to locally represent the surface of dielectric lens. The advantage of this approach is to analytically obtain the partial derivatives present in the MA equation. To validate the proposed methodology, numerical results are obtained by an iterative algorithm that allows the simultaneous control of the transmitted beam width and the power density amplitude in the far‐field region. To illustrate the synthesis effectiveness, two examples of lens antennas are simulated and analysed.

On partial uniqueness of complete non-compact Ricci flat metrics

Using techniques for Caccioppoli inequality, on a fairly general class of complete non-compact Kähler manifolds with sub-quadratic volume growth, we show uniqueness of bounded C 1 , 1 C^{1,1} solution to Monge-Ampere equation. This does not a priori require any decay of the solution.

Intersection of (1,1)-currents and the domain of definition of the Monge-Ampere operator

We study the Monge-Ampere operator $u \mapsto (\text{dd}^c u)^n$ and compare it with Dinh-Sibony's intersection product defined via density currents. We show that if $u$ is a plurisubharmonic function belonging to the Blocki-Cegrell class, then the Dinh-Sibony $n$-fold self-product of $\text{dd}^c u$ exists and coincides with $(\text{dd}^c u)^n$.

Discrete Aleksandrov solutions of the Monge-Ampere equation

We make two relaxations of the Oliker-Prussner method for the Dirichlet problem for the Monge-Ampere equation. First we relax the convexity requirement and consider mesh functions which are only discrete convex. The second relaxation consists in using a finite stencil. The discrete nonlinear equations are solved with a damped Newton's method. We give two proofs of convergence of the resulting scheme for right-hand side a density, on domains which are convex and not necessarily strictly convex, under the assumption that the boundary data has a continuous convex extension. The first proof is based on the notion of Aleksandrov solution while the second uses viscosity solutions. For more information see https://ejde.math.txstate.edu/conf-proc/26/a2/abstr.html

Asymptotic mean value formulas for parabolic nonlinear equations

In this paper we characterize viscosity solutions to nonlinear parabolic equations (including parabolic Monge-Ampere equations) by asymptotic mean value formulas. Our asymptotic mean value formulas can be interpreted from a probabilistic point of view in terms of Dynamic Programming Principles for certain two-player, zero-sum games.

Dimension reduction through gamma convergence for general prestrained thin elastic sheets

We study thin films with residual strain by analyzing the Gamma -limit of non-Euclidean elastic energy functionals as the material’s thickness tends to 0. We begin by extending prior results (Bhattacharya et al. in Arch Ration Mech Anal 228: 143–181, 2016); (Agostiniani et al. in ESAIM Control Opt Calculus Var 25: 24, 2019); (Lewicka and Lucic in Commun Pure Appl Math 73: 1880–1932, 2018); (Schmidt in J de Mathématiques Pures et Appliquées 88: 107–122, 2007) , to a wider class of films, whose prestrain depends on both the midplate and the transversal variables. The ansatz for our Gamma -convergence result uses a specific type of wrinkling, which is built on exotic solutions to the Monge-Ampere equation, constructed via convex integration (Lewicka and Pakzad in Anal PDE 10: 695–727, 2017). We show that the expression for our Gamma -limit has a natural interpretation in terms of the orthogonal projection of the residual strain onto a suitable subspace. We also show that some type of wrinkling phenomenon is necessary to match the lower bound of the Gamma -limit in certain circumstances. These results all assume a prestrain of the same order as the thickness; we also discuss why it is natural to focus on that regime by considering what can happen when the prestrain is larger.

Numerical methods for the hyperbolic Monge-Ampère equation based on the method of characteristics

We present three alternative derivations of the method of characteristics (MOC) for a second order nonlinear hyperbolic partial differential equation (PDE) in two independent variables. The MOC gives rise to two mutually coupled systems of ordinary differential equations (ODEs). As a special case we consider the Monge–Ampère (MA) equation, for which we present a general method of determining the location and number of required boundary conditions. We solve the systems of ODEs using explicit one-step methods (Euler, Runge-Kutta) and spline interpolation. Reformulation of the Monge–Ampère equation as an integral equation yields via its residual a proxy for the error of the numerical solution. Numerical examples demonstrate the performance and convergence of the methods.

Global pluripotential theory over a trivially valued field

We develop global pluripotential theory in the setting of Berkovich geometry over a trivially valued field. Specifically, we define and study functions and measures of finite energy and the non-Archimedean Monge-Ampere operator on any (possibly reducible) projective variety. We also investigate the topology of the space of valuations of linear growth, and the behavior of psh functions thereon.

Geometric theory of Weyl structures

Given a parabolic geometry on a smooth manifold [Formula: see text], we study a natural affine bundle [Formula: see text], whose smooth sections can be identified with Weyl structures for the geometry. We show that the initial parabolic geometry defines a reductive Cartan geometry on [Formula: see text], which induces an almost bi-Lagrangian structure on [Formula: see text] and a compatible linear connection on [Formula: see text]. We prove that the split-signature metric given by the almost bi-Lagrangian structure is Einstein with nonzero scalar curvature, provided that the parabolic geometry is torsion-free and [Formula: see text]-graded. We proceed to study Weyl structures via the submanifold geometry of the image of the corresponding section in [Formula: see text]. For Weyl structures satisfying appropriate non-degeneracy conditions, we derive a universal formula for the second fundamental form of this image. We also show that for locally flat projective structures, this has close relations to solutions of a projectively invariant Monge–Ampere equation and thus to properly convex projective structures.

Stability and Hölder regularity of solutions to complex Monge–Ampère equations on compact Hermitian manifolds

Let $(X,\omega)$ be a compact Hermitian manifold. We establish a stability result for solutions to complex Monge-Ampere equations with right-hand side in $L^p$, $p>1$. Using this we prove that the solutions are Holder continuous with the same exponent as in the Kahler case \cite{DDGKPZ14}. Our techniques also apply to the setting of big cohomology classes on compact Kahler manifolds.

Pinsker inequalities and related Monge-Ampere equations for log-concave functions

In this paper we further develop the theory of f-divergences for log-concave functions and their related inequalities. We establish Pinsker inequalities and new affine invariant entropy inequalities. We obtain new inequalities on functional affine surface area and lower and upper bounds for the Kullback-Leibler divergence in terms of functional affine surface area. The functional inequalities lead to new inequalities for L_p-affine surface areas for convex bodies.

Optimal System and Exact Solutions of Monge-Ampere Equation

A solution that remains unchanged when transformed under Lie group of point symmetries of the differential equation is an invariant solution of the differential equation. Optimal system of Lie group of point symmetry generators provide all possible invariant solutions of differential equation. Here, using optimal system of Lie point symmetry generators group invariant solutions are obtained. Using these solutions, exact solutions of non-homogeneous Monge-Ampere equation have been presented here.

Singular Monge-Ampere equations over convex domains

In this article we are interested in the Dirichlet problem for a class of singular Monge-Ampere equations over convex domains being either bounded or unbounded. By constructing a family of sub-solutions, we prove the existence and global Holderestimates of convex solutions to the problem over convex domains. The global regularity provided essentially depends on the convexity of the domain.
 For more information see https://ejde.math.txstate.edu/Volumes/2021/86/abstr.html

Fractional convexity

We introduce a notion of fractional convexity that extends naturally the usual notion of convexity in the Euclidean space to a fractional setting. With this notion of fractional convexity, we study the fractional convex envelope inside a domain of an exterior datum (the largest possible fractional convex function inside the domain that is below the datum outside) and show that the fractional convex envelope is characterized as a viscosity solution to a non-local equation that is given by the infimum among all possible directions of the 1-dimensional fractional laplacian. For this equation we prove existence, uniqueness and a comparison principle (in the framework of viscosity solutions). In addition, we find that solutions to the equation for the convex envelope are related to solutions to the fractional Monge–Ampere equation.

Resolution \xe0 la Kronheimer of mathbb {C}^3/Gamma singularities and the Monge\u2013Amp\xe8re equation for Ricci-flat K\xe4hler metrics in view of D3-brane solutions of supergravity

In this paper, we analyze the relevance of the generalized Kronheimer construction for the gauge/gravity correspondence. We begin with the general structure of D3-brane solutions of type IIB supergravity on smooth manifolds Y^Gamma that are supposed to be the crepant resolution of quotient singularities mathbb {C}^3/Gamma with Gamma a finite subgroup of SU(3). We emphasize that nontrivial 3-form fluxes require the existence of imaginary self-dual harmonic forms omega ^{2,1}. Although excluded in the classical Kronheimer construction, they may be reintroduced by means of mass deformations. Next we concentrate on the other essential item for the D3-brane construction, namely, the existence of a Ricci-flat metric on Y^Gamma . We study the issue of Ricci-flat Kähler metrics on such resolutions Y^Gamma , with particular attention to the case Gamma =mathbb {Z}_4. We advance the conjecture that on the exceptional divisor of Y^Gamma the Kronheimer Kähler metric and the Ricci-flat one, that is locally flat at infinity, coincide. The conjecture is shown to be true in the case of the Ricci-flat metric on mathrm{tot} K_{{mathbb {W}P}[112]} that we construct, i.e., the total space of the canonical bundle of the weighted projective space {mathbb {W}P}[112], which is a partial resolution of mathbb {C}^3/mathbb {Z}_4. For the full resolution, we have Y^{mathbb {Z}_4}={text {tot}} K_{mathbb {F}_{2}}, where mathbb {F}_2 is the second Hirzebruch surface. We try to extend the proof of the conjecture to this case using the one-parameter Kähler metric on mathbb {F}_2 produced by the Kronheimer construction as initial datum in a Monge–Ampère (MA) equation. We exhibit three formulations of this MA equation, one in terms of the Kähler potential, the other two in terms of the symplectic potential but with two different choices of the variables. In both cases, one can establish a series solution in powers of the variable along the fibers of the canonical bundle. The main property of the MA equation is that it does not impose any condition on the initial geometry of the exceptional divisor, rather it uniquely determines all the subsequent terms as local functionals of this initial datum. Although a formal proof is still missing, numerical and analytical results support the conjecture. As a by-product of our investigation, we have identified some new properties of this type of MA equations that we believe to be so far unknown.