On Extremal Permutations AvoidingωN = NN − 1 . . . 1

Dedicated to the memory of Marcel–Paul Schützenberger Cet article présente une étude des permutations qui évitent le motif de la permutation maximaleω_N = NN − 1 . . . 1. Après avoir donné les définitions classiques, nous montrons que l’ensemble de ces permutations est un idéal pour l’ordre de Bruhat faible et faisons l’étude de ses éléments maximaux. Nous exhibons alors un algorithme pour calculer ces éléments et nous montrons que ceux-ci peuvent être obtenus à partir d’un automate. Nous terminons en donnant des estimations asymptotiques de leur nombre. This paper presents a study of permutations avoiding the patternω_N = NN − 1 . . . 1. After recalling the basic definitions, we prove that this set of permutations is an ideal for the weak Bruhat order and begin the study of its maximal elements. We then present an algorithm that generates these elements and find out that they can be obtained from an automaton. Finally, we give some asymptotics about their number.     

On the solution of Dirichlet’s problem of the complex Monge–Ampère equation for the Cartan–Hartogs domain of the first type

The complex Monge–Ampère equation is a nonlinear equation with high degree; therefore getting its solution is very difficult. In the present paper how to get the solution of Dirichlet’s problem of the complex Monge–Ampère equation on the Cartan–Hartogs domain of the first type is discussed, using an analytic method. Firstly, the complex Monge–Ampère equation is reduced to a nonlinear ordinary differential equation, then the solution of Dirichlet’s problem of the complex Monge–Ampère equation is reduced to the solution of a two-point boundary value problem for a nonlinear second-order ordinary differential equation. Secondly, the solution of Dirichlet’s problem is given as a semi-explicit formula, and in a special case the exact solution is obtained. These results may be helpful for a numerical method approach to Dirichlet’s problem of the complex Monge–Ampère equation on the Cartan–Hartogs domain of the first type.     

The space of solutions to the Hessian one equation in the finitely punctured plane

We construct the space of solutions to the elliptic Monge–Ampère equation det(D²)=1 in the plane with n points removed. We show that, modulo equiaffine transformations and for n>1, this space can be seen as an open subset of , where the coordinates are described by the conformal equivalence classes of once punctured bounded domains in of connectivity n−1. This approach actually provides a constructive procedure that recovers all such solutions to the Monge–Ampère equation, and generalizes a theorem by K. Jörgens.     

New isoperimetric estimates for solutions to Monge–Ampère equations

We prove some sharp estimates for solutions to Dirichlet problems relative to Monge–Ampère equations. Among them we show that the eigenvalue of the Dirichlet problem, when computed on convex domains with fixed measure, is maximal on ellipsoids. This result falls in the class of affine isoperimetric inequalities and shows that the eigenvalue of the Monge–Ampère operator behaves just the contrary of the first eigenvalue of the Laplace operator.     

Existence and estimates of solutions to a singular Dirichlet problem for the Monge–Ampère equation

Given a strictly convex, smooth, and bounded domain Ω in we establish the existence of a negative convex solution in with zero boundary value to the singular Monge–Ampère equation det(D²u)=p(x)g(−u). An associated Dirichlet problem will be employed to provide a necessary and sufficient condition for the solvability of the singular boundary value problem. Estimates of solutions will also be given and regularity of solutions will be deduced from the estimates.     

On the behavior of strictly plurisubharmonic functions near real hypersurfaces

We describe the behavior of certain strictly plurisubharmonic functions near some real hypersurfaces in ℂ ⁿ , n≥3. Given a hypersurface we study continuous plurisubharmonic functions which are zero on the hypersurface and have Monge–Ampère mass greater than one in a one-sided neighborhood of the hypersurface. If we can find complex curves which have sufficiently high contact order with the hypersurface then the plurisubharmonic functions we study cannot be globally Lipschitz in the one-sided neighborhood.     

A Geometric approximation to the euler equations: the Vlasov–Monge–Ampère system

This paper studies the Vlasov–Monge–Ampère system (VMA), a fully non-linear version of the Vlasov–Poisson system (VP) where the (real) Monge–Ampère equation det ²/x_i x_j = substitutes for the usual Poisson equation. This system can be derived as a geometric approximation of the Euler equations of incompressible fluid mechanics in the spirit of Arnold and Ebin. Global existence of weak solutions and local existence of smooth solutions are obtained. Links between the VMA system, the VP system and the Euler equations are established through rigorous asymptotic analysis.     

The complex Monge–Ampère equation with infinite boundary value

In this article we consider the complex Monge–Ampère equation with infinite boundary value in bounded pseudoconvex domains. We prove the existence of strictly plurisubharmonic solution to the problem in convex domains under suitable growth conditions. We also obtain, for general pseudoconvex domains, some nonexistence results which show that these growth conditions are nearly optimal.     

Hypersurfaces with Prescribed Affine Gauss–Kronecker Curvature

We extend a procedure for solving particular fourth order PDEs by splitting them into two linked second order Monge–Ampère equations. We use this for the global study of Blaschke hypersurfaces with prescribed Gauss–Kronecker curvature.     

Monge–Ampère of Pac-Man

We show that the Monge–Ampère density of the extremal function $$V_P$$ for a non-convex Pac-Man set $$P\subset {{\mathbb {R}}}^2$$ tends to a finite limit as we approach the vertex p of P along lines but with a value that may vary with the line. On the other hand, along a tangential approach to p, the Monge–Ampère density becomes unbounded. This partially mimics the behavior of the Monge–Ampère density of the union of two quarter disks S of Sigurdsson and Snaebjarnarson (Ann Pol Math 123:481–504, 2019). We also recover their formula for $$V_S$$ by elementary methods.     

Vector spaces of delta-plurisubharmonic functions and extensions of the complex Monge–Ampère operator

In this paper we shall consider two types of vector ordering on the vector space of differences of negative plurisubharmonic functions, and the problem whether it is possible to construct supremum and infimum. Then we consider two different approaches to define the complex Monge–Ampère operator on these vector spaces, and we solve some Dirichlet problems. We end this paper by stating and discussing some open problems.     

A local regularity of the complex Monge–Ampère equation

We prove a local regularity (and a corresponding a priori estimate) for plurisubharmonic solutions of the nondegenerate complex Monge–Ampère equation assuming that their W ^{2, p} -norm is under control for some p > n(n − 1). This condition is optimal. We use in particular some methods developed by Trudinger and an estimate for the complex Monge–Ampère equation due to Kołodziej.     

B-splines and Hermite–Padé approximants to the exponential function

This paper is the continuation of a work initiated in [P. Sablonnière, An algorithm for the computation of Hermite–Padé approximations to the exponential function: divided differences and Hermite–Padé forms. Numer. Algorithms 33 (2003) 443–452] about the computation of Hermite–Padé forms (HPF) and associated Hermite–Padé approximants (HPA) to the exponential function. We present an alternative algorithm for their computation, based on the representation of HPF in terms of integral remainders with B-splines as Peano kernels. Using the good properties of discrete B-splines, this algorithm gives rise to a great variety of representations of HPF of higher orders in terms of HPF of lower orders, and in particular of classical Padé forms. We give some examples illustrating this algorithm, in particular, another way of constructing quadratic HPF already described by different authors. Finally, we briefly study a family of cubic HPF.     

Monge–Ampère measures on pluripolar sets

In this article we solve the complex Monge–Ampère problem for measures with large singular part. This result generalizes classical results by Demailly, Lelong and Lempert a.o., who considered singular parts carried on discrete sets. By using our result we obtain a generalization of Ko?odziej's subsolution theorem. More precisely, we prove that if a non-negative Borel measure is dominated by a complex Monge–Ampère measure, then it is a complex Monge–Ampère measure.     

Kähler–Ricci solitons on compact complex manifolds with C 1(M) > 0

In this paper, we discuss the relation between the existence of Kähler–Ricci solitons and a certain functional associated to some complex Monge–Ampère equation on compact complex manifolds with positive first Chern class. In particular, we obtain a strong inequality of Moser–Trudinger type on a compact complex manifold admitting a Kähler–Ricci soliton.Received: October 2004 Revised: February 2005 Accepted: February 2005     

Interior gradient estimates for solutions to the linearized Monge–Ampère equation

Let ? be a convex function on a convex domain Ω⊂Rn, n?1. The corresponding linearized Monge–Ampère equation istrace(ΦD²u)=f, where is the matrix of cofactors of D²?. We establish interior Hölder estimates for derivatives of solutions to such equation when the function f on the right-hand side belongs to Lp(Ω) for some p>n. The function ? is assumed to be such that with ?=0 on ∂Ω and the Monge–Ampère measure is given by a density g∈C(Ω) which is bounded away from zero and infinity.     

Gevrey regularity of subelliptic Monge–Ampère equations in the plane

In this paper, we establish the Gevrey regularity of solutions for a class of degenerate Monge–Ampère equations in the plane. Under the assumptions that one principal entry of the Hessian is strictly positive and the coefficient of the equation is degenerate with appropriately finite type degeneracy, we prove that the solution of the degenerate Monge–Ampère equation will be smooth in Gevrey classes.     

The Cauchy problem for the homogeneous Monge–Ampère equation, II. Legendre transform

We continue our study of the Cauchy problem for the homogeneous (real and complex) Monge–Ampère equation (HRMA/HCMA). In the prequel (Y.A. Rubinstein and S. Zelditch [27]) a quantum mechanical approach for solving the HCMA was developed, and was shown to coincide with the well-known Legendre transform approach in the case of the HRMA. In this article—that uses tools of convex analysis and can be read independently—we prove that the candidate solution produced by these methods ceases to solve the HRMA, even in a weak sense, as soon as it ceases to be differentiable. At the same time, we show that it does solve the equation on its dense regular locus, and we derive an explicit a priori upper bound on its Monge–Ampère mass. The technique involves studying regularity of Legendre transforms of families of non-convex functions.     

Uniform decay in a Timoshenko-type system with past history

Fernández Sare and Rivera [H.D. Fernández Sare, J.E. Muñoz Rivera, Stability of Timoshenko systems with past history, J. Math. Anal. Appl. 339 (1) (2008) 482–502] considered the following Timoshenko-type system

ρ₁φ_tt−K(φ_x+ψ)_x=0,