Grid analogs of two functions harmonic on the plane with a cut

For the Laplace equation in an unbounded domain (in the first quadrant, upper half-plane, plane with a cut), the Dirichlet and Neumann problems whose solutions are the imaginary and real parts of the complex function z ²lnz, respectively, are considered. Both problems are approximated on a square grid using the classic five-point difference scheme. The grid Fourier transform is applied to represent the solutions to the aforementioned grid problems in an integral form and obtain their asymptotic decompositions. It follows from the results that the accuracy of these grid solutions in the L _∞ ^h norm is O(h ²), where h is the grid step.     

A refinement of the existence and uniqueness theorem for the Vlasov-Poisson system

In a recent paper Horst shows that if a classical solution to the Vlasov-Poisson system ceases to exist then at this point in time not only does velocity support become unbounded but support in position space becomes unbounded also (assuming compactly supported initial data). In the present paper we formulate this result another way and give a different proof. It is shown that an a priori bound on the support of solutions in position space leads to an a priori bound on the support in velocity space and hence existence and uniqueness of solutions. Thus a necessary and sufficient condition for solvability is that the system admit an a priori bound on the support in position space alone. This gives a refinement of Wollman (J. Math. Anal. Appl., 90 1982, p. 141, Theorem 2.1).     

Normal solvability of linear elliptic problemsRésolubilité normale des problèmes elliptiques linéaires

The paper is devoted to general linear elliptic problems in Hölder spaces. We consider unbounded domains and define limiting problems at infinity. We give a necessary and sufficient condition of normal solvability through uniqueness of solutions of limiting problems. We study a structure of spaces dual to Hölder spaces and specify the subspace of functionals, which provide the condition of normal solvability. This allows us to prove that for Fredholm operators all limiting operators are invertible. To cite this article: V. Volpert, A. Volpert, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 457–462.     

Explosion et normes Lp pour l'équation des ondes nonlinéaire cubiqueBlow-up and Lp norms for the cubic nonlinear wave equation

We find blow-up solutions of nonlinear wave equations with cubic nonlinearity, in any number of space dimensions, and study the asymptotic behavior of their L^p norms and “energy”. The L^p norm blows up if the blow-up surface has an interior non-degenerate minimum and p?n/2. For less smooth right-hand sides, and 0<ε<1, we give examples for which the L^p norm blows up if p?n/(1+ε); their Cauchy data are unbounded, but blow-up is not instantaneous. Applications to nonlinear optics are briefly outlined. To cite this article: G. Cabart, S. Kichenassamy, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 903–908.     

On a viscous Hamilton-Jacobi equation with an unbounded potential term

We present comparison, uniqueness and existence results for unbounded solutions of a viscous Hamilton-Jacobi or eikonal equation. The equation includes an unbounded potential term V(x) subject to a quadratic upper bound. The results are obtained through a tailor-made change of variables in combination with the Hopf-Cole transformation. An integral representation formula for the solution of the Cauchy problem is derived in the case where V(x)=ω²|x|²/2.     

Radiation condition and uniqueness for the outgoing elastic wave in a half-plane with free boundary

In this Note we deduce an explicit Sommerfeld-type radiation condition which is convenient to prove the uniqueness for the time-harmonic outgoing wave problem in an isotropic elastic half-plane with free boundary condition. The expression is obtained from a rigorous asymptotic analysis of the associated Green's function. The main difficulty is that the free boundary condition allows the propagation of a Rayleigh wave which cannot be neglected in the far field expansion. We also give the existence result for this problem. To cite this article: M. Durán et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

A singular Monge-Ampère equation on unbounded domains

In this paper, we study the Dirichlet problem for a singular Monge-Amp`ere type equation on unbounded domains. For a few special kinds of unbounded convex domains, we find the explicit formulas of the solutions to the problem. For general unbounded convex domain ?, we prove the existence for solutions to the problem in the space C∞(?) ∩ C(?). We also obtain the local C1/2-estimate up to the ?? and the estimate for the lower bound of the solutions.     

Steady flows of Jeffrey–Hamel type from the half-plane into an infinite channel. 2. Linearization on a symmetric solution

The steady flow of a Navier–Stokes fluid is analysed in a two-dimensional asymmetric unbounded domain $\text{Ω}$, having two outlets to infinity, namely a half-plane K and a semi-infinite channel Π₋. Assuming that $\text{Ω}$ differs from a symmetric domain $\text{Ω}$ only by a small perturbation, we show the existence of a unique solution to the Navier–Stokes system in $\text{Ω}$. The solution is obtained as a perturbation of the symmetric solution and, at large distances in K, it takes the Jeffrey–Hamel form. Curiously, our results are valid only if the flux Φ, besides being small, is directed from the half-plane towards the semi-infinite channel, i.e. Φ is negative.The main ingredients in our proofs are estimates in weighted spaces with detached asymptotics and the study of a model problem resulting from the linearization around the symmetric solution which, for non-zero flux, leads, in contrast to the linearization around the zero solution, to the absence of compatibility conditions for the convective term and, for Φ<0, to the domination of nonlinear terms by the linear ones. We also provide some explicit examples of the domain perturbation.     

An exact bounded PML for the Helmholtz equation

We study the Helmholtz equation with a Sommerfeld radiation condition in an unbounded domain. We prove the existence of an exact bounded perfectly matched layer (PML) for this problem, in the sense that we recover the exact solution in the physical domain by choosing a singular PML function in a bounded domain. We approximate the solution for the PML problem using a standard finite element method and assess its performance through numerical tests. To cite this article: A. Bermúdez et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Hyperbolic Equations with Growing Coefficients in Unbounded Domains

Let Ω ? ? ⁿ , n?≥?2, be an unbounded domain with a smooth (possibly noncompact) star-shaped boundary Γ. For the first mixed problem for a hyperbolic equation with an unbounded coefficient with power growth at infinity, the large-time behavior of the solutions is studied. Estimates for the resolvent of the spectral problem are obtained for various values of the parameters.     

Polynomials with the half-plane property and matroid theory

A polynomial f is said to have the half-plane property if there is an open half-plane H⊂C, whose boundary contains the origin, such that f is non-zero whenever all the variables are in H. This paper answers several open questions relating multivariate polynomials with the half-plane property to matroid theory.

(1)

We prove that the support of a multivariate polynomial with the half-plane property is a jump system. This answers an open question posed by Choe, Oxley, Sokal and Wagner and generalizes their recent result claiming that the same is true whenever the polynomial is also homogeneous.

(2)

We prove that a multivariate multi-affine polynomial f∈R[z₁,…,zn] has the half-plane property (with respect to the upper half-plane) if and only if

     

Exact travelling wave solutions of the generalized Bretherton equation

We search for exact travelling wave solutions of the generalized Bretherton equation for integer, greater than one, values of the exponent m of the nonlinear term by two methods: the truncated Painlevé expansion method and an algebraic method. We find periodic solutions for m=2 and m=5, to add to those already known for m=3; in all three cases these solutions exist for finite intervals of the wave velocity. We also find a “kink” shaped solitary wave for m=5 and a family of elementary unbounded solutions for arbitrary m.     

Proof complexity of intuitionistic implicational formulas

We study implicational formulas in the context of proof complexity of intuitionistic propositional logic (IPC). On the one hand, we give an efficient transformation of tautologies to implicational tautologies that preserves the lengths of intuitionistic extended Frege (EF) or substitution Frege (SF) proofs up to a polynomial. On the other hand, EF proofs in the implicational fragment of IPC polynomially simulate full intuitionistic logic for implicational tautologies. The results also apply to other fragments of other superintuitionistic logics under certain conditions.In particular, the exponential lower bounds on the length of intuitionistic EF proofs by Hrube? (2007), generalized to exponential separation between EF and SF systems in superintuitionistic logics of unbounded branching by Je?ábek (2009), can be realized by implicational tautologies.     

Bimodules de Kasparov non bornés équivariants pour les groupoïdes topologiques localement compacts

We study the unbounded KK-theory of S. Baaj and P. Julg in the equivariant framework concerning the action of locally compact groups and groupoids, and give some geometrical examples. To cite this article: F. Pierrot, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

On the asymptotic behaviour of elliptic problems with periodic data

We study the asymptotic behaviour of the solution of elliptic problems with periodic data when the size of the domain on which the problem is set becomes unbounded. To cite this article: M. Chipot, Y. Xie, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Steady flow for incompressible fluids in domains with unbounded curved channels

We give an overview on the solution of the stationary Navier-Stokes equations for non newtonian incompressible fluids established by G. Dias and M.M. Santos (Steady flow for shear thickening fluids with arbitrary fluxes, J. Differential Equations 252 (2012), no. 6, 3873-3898), propose a definition for domains with unbounded curved channels which encompasses domains with an unbounded boundary, domains with nozzles, and domains with a boundary being a punctured surface, and argue on the existence of steady flowfor incompressible fluids with arbitrary fluxes in such domains.     

Weighted Sobolev embedding with unbounded and decaying radial potentials

We prove embedding results of weighted W1,p(RN) spaces of radially symmetric functions. The results then are used to obtain ground and bound state solutions of quasilinear equations with unbounded or decaying radial potentials.     

Quasilinear scalar field equations with competing potentials

We are concerned with the existence and non-existence of nontrivial weak solutions for a class of quasilinear scalar field equations in RN driven by competing nonlinearities with general potentials which can be unbounded or decaying to zero as |x|→+∞. Furthermore, the existence of ground states and/or bound states is considered.     

On the univalence of derivatives of functions which are univalent in angular domains

We consider functions f that are univalent in a plane angular domain of angle απ, 0 < α ≤ 2. It is proved that there exists a natural number k depending only on α such that the kth derivatives f ^(k) of these functions cannot be univalent in this angle. We find the least of the possible values of for k. As a consequence, we obtain an answer to the question posed by Kir’yatskii: if f is univalent in the half-plane, then its fourth derivative cannot be univalent in this half-plane.