Some examples for the Poincaré and Painlevé problems

In 1891, Poincaré started a series of three papers in which he tried to answer the following question (cf. [21-23]): “Is it possible to decide if an algebraic differential equation in two variables is algebraically integrable?” (in the sense that it has a rational first integral). More or less at the same time P. Painlevé asked the following question: “Is it possible to recognize the genus of the general solution of an algebraic differential equation in two variables which has a rational first integral?”. In this paper we give examples of one-parameter families which show that both problems have a negative answer. With some of the families we can also answer a question posed by M. Brunella in [5].     

Local “superlinearity” and “sublinearity” for the p-Laplacian

We study the existence, nonexistence and multiplicity of positive solutions for a family of problems −Δpu=fλ(x,u), , where Ω is a bounded domain in RN, N>p, and λ>0 is a parameter. The family we consider includes the well-known nonlinearities of Ambrosetti-Brezis-Cerami type in a more general form, namely λa(x)uq+b(x)ur, where 0?q<p−1<r?p^∗−1. Here the coefficient a(x) is assumed to be nonnegative but b(x) is allowed to change sign, even in the critical case. Preliminary results of independent interest include the extension to the p-Laplacian context of the Brezis-Nirenberg result on local minimization in and , a C1,α estimate for equations of the form −Δpu=h(x,u) with h of critical growth, a strong comparison result for the p-Laplacian, and a variational approach to the method of upper-lower solutions for the p-Laplacian.     

Sharp Békollé estimates for the Bergman projection

We prove sharp estimates for the Bergman projection in weighted Bergman spaces in terms of the Békollé constant. Our main tools are a dyadic model dominating the operator and an adaptation of a method of Cruz-Uribe, Martell and Pérez.     

On the threshold for the Maker‐Breaker H‐game

We study the Maker‐Breaker H‐game played on the edge set of the random graph . In this game two players, Maker and Breaker, alternately claim unclaimed edges of , until all edges are claimed. Maker wins if he claims all edges of a copy of a fixed graph H; Breaker wins otherwise. In this paper we show that, with the exception of trees and triangles, the threshold for an H‐game is given by the threshold of the corresponding Ramsey property of with respect to the graph H. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 49, 558–578, 2016     

The “hot spots” conjecture for the Sierpinski gasket

There are many works on the “hot spots” conjecture for domains in Euclidean space since the conjecture was posed by J. Rauch in 1974. In this paper, using spectral decimation, we prove that the conjecture holds on the Sierpinski gasket, i.e., every eigenfunction of the second-smallest eigenvalue of the Neumann Laplacian (introduced by J. Kigami) attains its maximum and minimum on the boundary.     

Blowup of solutions for the “bad” Boussinesq-type equation

The paper studies the blowup of solutions to the initial boundary value problem for the “bad” Boussinesq-type equation u_tt−u_xx−bu_xxxx=σ(u)_xx, where b>0 is a real number and σ(s) is a given nonlinear function. By virtue of the energy method and the Fourier transform method, respectively, it proves that under certain assumptions on σ(s) and initial data, the generalized solutions of the above-mentioned problem blow up in finite time. And a few examples are shown, especially for the “bad” Boussinesq equation, two examples of blowup of solutions are obtained numerically.     

A “discretization” technique for the solution of ODEs

A functional-analytic technique was developed in the past for the establishment of unique solutions of ODEs in H₂(D) and H₁(D) and of difference equations in ?₂ and ?₁. This technique is based on two isomorphisms between the involved spaces. In this paper, the two isomorphisms are combined in order to find discrete equivalent counterparts of ODEs, so as to obtain eventually the solution of the ODEs under consideration. As an application, the Duffing equation and the Lorenz system are studied. The results are compared with numerical ones obtained using the 4th order Runge-Kutta method. The advantages of the present method are that, it is accurate, the only errors involved are the round-off errors, it does not depend on the grid used and the obtained solution is proved to be unique.     

Equidistribution for meromorphic transforms and the dd<Superscript>c</Superscript>-method

In this paper we give an introduction to the notion of meromorphic transform. We describe some equidistribution problems and their solution, using the dd^c-method. In particular, we give some statistical properties of the equilibrium measure for meromorphic maps on compact Kähler manifolds: K-mixing, exponential decay of correlations and central limit theorem.     

Paley‐Wiener‐Type theorems for the Clifford‐Fourier transform

In the framework of Clifford analysis, we consider the Paley‐Wiener type theorems for a generalized Clifford‐Fourier transform. This Clifford‐Fourier transform is given by a similar operator exponential as the classical Fourier transform but containing generators of Lie superalgebra.     

Non-linear stability for the Vlasov–Poisson system—the energy-Casimir method

The problem of stability of stationary solutions of the Vlasov–Poisson system has received a lot of attention in the physics literature, both in the stellar dynamics and the plasma physics cases. The energy-Casimir method has been used to prove non-linear stability for various conservative systems, but no rigorous application to the Vlasov–Poisson system has been given yet. We employ this method to prove non-linear stability of stationary solutions for the plasma physics case in three geometrically different settings.     

ℋ︁‐matrices for the convection‐diffusion equation

Hierarchical matrices (ℋ︁‐matrices) provide a technique for the sparse approximation of large, fully populated matrices. This technique has been shown to be applicable to stiffness matrices arising in boundary element method applications where the kernel function displays certain smoothness properties. The error estimates for an approximation of the kernel function by a separable function can be carried over directly to error estimates for an approximation of the stiffness matrix by an ℋ︁‐matrix, using a certain standard partitioning and admissibility condition for matrix blocks. Similarly, ℋ︁‐matrix techniques can be applied in the finite element context where it is the inverse of the stiffness matrix that is fully populated. Here one needs a separable approximation of Green's function of the underlying boundary value problem in order to prove approximability by matrix blocks of low rank. Unfortunately, Green's function for the convection‐diffusion equation does not satisfy the required smoothness properties, hence prohibiting a straightforward generalization of the separable approximation through Taylor polynomials. We will use Green's function to motivate a modification in the (hierarchical) partitioning of the index set and as a consequence the resulting hierarchy of block partitionings as well as the admissibility condition. We will illustrate the effect of the proposed modifications by numerical results.     

A convolution for the Hankel–Kontorovich–Lebedev transformation

The analysis of a kernel involving the product of three Bessel functions motivates the introduction of the translation operator and the convolution associated to the Hankel–Kontorovich–Lebedev tranformation, first in a classical framework, and then in certain spaces of generalized functions. The main properties of this convolution are investigated, the more important operational rules are obtained and some applications are shown. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Well‐posedness for the incompressible magneto‐hydrodynamic system

This paper is concerned with well‐posedness of the incompressible magneto‐hydrodynamics (MHD) system. In particular, we prove the existence of a global mild solution in BMO^?1 for small data which is also unique in the space C([0, ∞); BMO^?1). We also establish the existence of a local mild solution in bmo^?1 for small data and its uniqueness in C([0, T); bmo^?1). In establishing our results an important role is played by the continuity of the bilinear form which was proved previously by Kock and Tataru. In this paper, we give a new proof of this result by using the weighted L^p‐boundedness of the maximal function. Copyright © 2006 John Wiley & Sons, Ltd.     

On the Phragmén-Lindelöf principle for second-order elliptic equations

This paper is concerned with the maximum principle for subsolutions of second-order elliptic equations in non-divergence form in unbounded domains. Eventually the zero-order term can change sign and the involved functions can be unbounded at infinity with an admissible growth depending on the geometric properties of the domain. Following Gilbarg and Hopf, we also show a Phragmén-Lindelöf principle in angular sectors and give an example of an interesting field of application to nonlinear equations, deriving comparison principles for quasi-linear operators.     

A uniform estimate for the MMOC for two‐dimensional advection‐diffusion equations

We prove an optimal‐order error estimate in a weighted energy norm for the modified method of characteristics (MMOC) and the modified method of characteristics with adjusted advection (MMOCAA) for two‐dimensional time‐dependent advection‐diffusion equations, in the sense that the generic constants in the estimates depend on certain Sobolev norms of the true solution but not on the scaling diffusion parameter ε. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Methods for the numerical solution of the Benjamin‐Bona‐Mahony‐Burgers equation

L^∞‐error estimates for finite element for Galerkin solutions for the Benjamin‐Bona‐Mahony‐Burgers (BBMB) equation are considered. A priori bound and the semidiscrete Galerkin scheme are studied using appropriate projections. For fully discrete Galerkin schemes, we consider the backward Euler method and analyze the corresponding error estimates. For a second order accuracy in time, we propose a three‐level backward method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Blow‐up for the b‐family of equations

In this paper, we consider the b‐family of equations on the torus u _t ?u _{t x x} +(b + 1)u u _x =b u _x u _{x x} +u u _{x x x} , which for appropriate values of b reduces to well‐known models, such as the Camassa–Holm equation or the Degasperis–Procesi equation. We establish a local‐in‐space blow‐up criterion. Copyright © 2016 John Wiley & Sons, Ltd.