A system of elliptic equations arising in Chern-Simons field theory

We prove the existence of topological vortices in a relativistic self-dual Abelian Chern-Simons theory with two Higgs particles and two gauge fields through a study of a coupled system of two nonlinear elliptic equations over R². We present two approaches to prove existence of solutions on bounded domains: via minimization of an indefinite functional and via a fixed point argument. We then show that we may pass to the full R² limit from the bounded-domain solutions to obtain a topological solution in R².     

Quantization effects for Maxwell-Chern-Simons vortices

In this paper, we prove the quantization of potential energy of topological Maxwell-Chern-Simons vortices on R² when the gauge fields vanish. We also show the nonexistence of nontopological solutions.     

Analytic estimates and rigorous continuation for equilibria of higher-dimensional PDEs

In this paper we extend the ideas of the so-called validated continuation technique to the context of rigorously proving the existence of equilibria for partial differential equations defined on higher-dimensional spatial domains. For that effect we present a new set of general analytic estimates. These estimates are valid for any dimension and are used, together with rigorous computations, to construct a finite number of radii polynomials. These polynomials provide a computationally efficient method to prove, via a contraction argument, the existence and local uniqueness of solutions for a rather large class of nonlinear problems. We apply this technique to prove existence and local uniqueness of equilibrium solutions for the Cahn-Hilliard and the Swift-Hohenberg equations defined on two- and three-dimensional spatial domains.     

Multiple vortices for a self-dual CP(1) Maxwell-Chern-Simons model

We prove the existence of at least two doubly periodic vortex solutions for a self-dual CP(1) Maxwell-Chern-Simons model. To this end we analyze a system of two elliptic equations with exponential nonlinearities. Such a system is shown to be equivalent to a fourth-order elliptic equation admitting a variational structure. Tonia Ricciardi: Partially supported by the MIUR National Project Variational Methods and Nonlinear Differential Equations     

On the growth of meromorphic solutions of the Schwarzian differential equations

We study the properties of meromorphic solutions of the Schwarzian differential equations in the complex plane by using some techniques from the study of the class Wp. We find some upper bounds of the order of meromorphic solutions for some types of the Schwarzian differential equations. We also show that there are no wandering domains nor Baker domains for meromorphic solutions of certain Schwarzian differential equations.     

The Chern-Simons invariants of cone-manifolds with Whitehead link singular set

In the present article, we obtain some explicit integral formulas for the generalized Chern-Simons function I(W(α,β)) for Whitehead link cone-manifolds in the hyperbolic and spherical cases. We also give the Chern-Simons invariant for the Whitehead link orbifolds. We find a formula for the Chern-Simons invariant of n-fold coverings of the three-sphere branched over the Whitehead link.     

Global solvability for the Kirchhoff equations in exterior domains of dimension three

We consider the initial (boundary) value problem for the Kirchhoff equations in exterior domains or in the whole space of dimension three, and show that these problems admit time-global solutions, provided the norms of the initial data in the usual Sobolev spaces of appropriate order are sufficiently small. We obtain uniform estimates of the L¹(R) norms with respect to time variable at each point in the domain, of solutions of initial (boundary) value problem for the linear wave equations. We then show that the estimates above yield the unique global solvability for the Kirchhoff equations.     

Boundary Regularity for Viscosity Solutions of Fully Nonlinear Elliptic Equations

We provide regularity results at the boundary for continuous viscosity solutions to nonconvex fully nonlinear uniformly elliptic equations and inequalities in Euclidian domains. We show that (i) any solution of two sided inequalities with Pucci extremal operators is C ^{1, α} on the boundary; (ii) the solution of the Dirichlet problem for fully nonlinear uniformly elliptic equations is C ^{2, α} on the boundary; (iii) corresponding asymptotic expansions hold. This is an extension to viscosity solutions of the classical Krylov estimates for smooth solutions.     

Local and global uniform convergence for elliptic problems on varying domains

The aim of the paper is to prove optimal results on local and global uniform convergence of solutions to elliptic equations with Dirichlet boundary conditions on varying domains. We assume that the limit domain be stable in the sense of Keldyš [Amer. Math. Soc. Transl. 51 (1966) 1-73]. We further assume that the approaching domains satisfy a necessary condition in the inside of the limit domain, and only require L²-convergence outside. As a consequence, uniform and L²-convergence are the same in the trivial case of homogenisation of a perforated domain. We are also able to deal with certain cracking domains.     

On the Rate of Convergence of Difference Approximations for Uniformly Nondegenerate Elliptic Bellman’s Equations

We show that the rate of convergence of solutions of finite-difference approximations for uniformly elliptic Bellman’s equations is of order at least h ^2/3, where h is the mesh size. The equations are considered in smooth bounded domains.     

On the Domain Dependence of Solutions to the Compressible Navier-Stokes Equations of an Isothermal Fluid

The aim of this paper is to study the behaviour of the variational solutions to the Navier-Stokes equations describing viscous compressible isothermal fluids with nonlinear stress tensors in a sequence of domains $\{\varOmega_{n}\} _{n=1}^{\infty}$ . The sequence converges in sense of the Sobolev-Orlicz capacity to domain Ω. We prove that the solutions of the equations in Ω _n converge to a solution of the respective equations in Ω. Moreover, The result can be applied to generalization of the existence result.     

Existence locale et unicité d'écoulements de fluides viscoélastiques dans des domaines non bornés

In this Note, we present a result of local existence and uniqueness, for any initial data, of the solutions to the equations of viscoelastic fluids of Jeffreys type (differential constitutive law). The system of equations is supposed to be verified in an unbounded domain Ω ⊂ ℝ^N (N = 2 or 3)), uniformly regular. The difficulty comes essentially from the loss of compactness in the case of unbounded domains. To overcome this difficulty we use a local compactness method, which allows us to construct a sequence of solutions on subdomains ;inn whi_n which union covers Ω After that, we pass to the limit to define a solution over the whole domain. Finally we show the uniqueness of this solution in its class of regularity, by using an energy estimate.     

Existence and multiplicity of nodal solutions for Dirichlet problems in upper half strip with holes

In this paper, we consider the existence and multiplicity of nodal solutions of semilinear elliptic equations. We prove that a semilinear elliptic equation in large domains does not admit any least energy nodal (sign-changing) solution and in an upper half strip with m-holes has at least m² 2-nodal solutions.     

Navier-Stokes equations in 3D thin domains with Navier friction boundary condition

In this article we study the 3D Navier-Stokes equations with Navier friction boundary condition in thin domains. We prove the global existence of strong solutions to the 3D Navier-Stokes equations when the initial data and external forces are in large sets as the thickness of the domain is small. We generalize the techniques developed to study the 3D Navier-Stokes equations in thin domains, see [G. Raugel, G. Sell, Navier-Stokes equations on thin 3D domains I: Global attractors and global regularity of solutions, J. Amer. Math. Soc. 6 (1993) 503-568; G. Raugel, G. Sell, Navier-Stokes equations on thin 3D domains II: Global regularity of spatially periodic conditions, in: Nonlinear Partial Differential Equations and Their Application, College de France Seminar, vol. XI, Longman, Harlow, 1994, pp. 205-247; R. Temam, M. Ziane, Navier-Stokes equations in three-dimensional thin domains with various boundary conditions, Adv. Differential Equations 1 (1996) 499-546; R. Temam, M. Ziane, Navier-Stokes equations in thin spherical shells, in: Optimization Methods in Partial Differential Equations, in: Contemp. Math., vol. 209, Amer. Math. Soc., Providence, RI, 1996, pp. 281-314], to the Navier friction boundary condition by introducing a new average operator Mε in the thin direction according to the spectral decomposition of the Stokes operator Aε. Our analysis hinges on the refined investigation of the eigenvalue problem corresponding to the Stokes operator Aε with Navier friction boundary condition.     

Parabolic equations in time dependent Reifenberg domains

In this paper we define time dependent parabolic Reifenberg domains and study Lp estimates for weak solutions of uniformly parabolic equations in divergence form on these domains. The basic assumption is that the principal coefficients are of parabolic BMO space with small parabolic BMO seminorms. It is shown that Lp estimates hold for time dependent parabolic δ-Reifenberg domains.     

Real-parameter square-integrable solutions and the spectrum of differential operators

We continue to investigate the connection between the spectrum of self-adjoint ordinary differential operators with arbitrary deficiency index d and the number of linearly independent square-integrable solutions for real values of the spectral parameter λ. We show that if, for all λ in an open interval I, there are d linearly independent square-integrable solutions, then there is no continuous spectrum in I. This for any self-adjoint realization with boundary conditions which may be separated, coupled, or mixed. The proof is based on a new characterization of self-adjoint domains and on limit-point (LP) and limit-circle (LC) solutions established in an earlier paper.     

Schwarz symmetrization and comparison results for nonlinear elliptic equations and eigenvalue problems

We compare the distribution function and the maximum of solutions of nonlinear elliptic equations defined in general domains with solutions of similar problems defined in a ball using Schwarz symmetrization. As an application, we prove the existence and bound of solutions for some nonlinear equation. Moreover, for some nonlinear problems, we show that if the first p-eigenvalue of a domain is big, the supremum of a solution related to this domain is close to zero. For that we obtain L ^∞ estimates for solutions of nonlinear and eigenvalue problems in terms of other L ^p norms.     

Transparent boundary conditions for a class of boundary value problems in some ramified domains with a fractal boundary

We consider some boundary value problems in self-similar ramified domains, with Laplace and Helmholtz equations. We discuss transparent boundary conditions. These conditions permit computing the restriction of the solutions to domains obtained by stopping the geometric construction after a finite number of steps. To cite this article: Y. Achdou et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function

In this paper, we study the combined effect of concave and convex nonlinearities on the number of positive solutions for semilinear elliptic equations with a sign-changing weight function. With the help of the Nehari manifold, we prove that there are at least two positive solutions for Eq. (Eλ,f) in bounded domains.     

On multivariable encryption schemes based on simultaneous algebraic Riccati equations over finite fields

New multivariable asymmetric public-key encryption schemes based on the NP-complete problem of simultaneous algebraic Riccati equations over finite fields are suggested. We also provide a systematic way to describe any set of quadratic equations over any field, as a set of algebraic Riccati equations. This has the benefit of systematic algebraic crypt-analyzing any encryption scheme based on quadratic equations, to any possible vulnerable hidden structure, in view of the fact that the set of all solutions to any given single algebraic Riccati equation is fully described in terms of all the T-invariant subspaces of some restricted dimension, where T is the matrix of coefficients of the related algebraic Riccati equation.