The proof of the Lane-Emden conjecture in four space dimensions

We partially solve a well-known conjecture about the nonexistence of positive entire solutions to elliptic systems of Lane-Emden type when the pair of exponents lies below the critical Sobolev hyperbola. Up to now, the conjecture had been proved for radial solutions, or in n?3 space dimensions, or in certain subregions below the critical hyperbola for n?4. We here establish the conjecture in four space dimensions and we obtain a new region of nonexistence for n?5. Our proof is based on a delicate combination involving Rellich-Pohozaev type identities, a comparison property between components via the maximum principle, Sobolev and interpolation inequalities on Sn−1, and feedback and measure arguments. Such Liouville-type nonexistence results have many applications in the study of nonvariational elliptic systems.     

Existence of pure Nash equilibria in discontinuous and non quasiconcave games

In a recent but well known paper, Reny has proved the existence of Nash equilibria for compact and quasiconcave games, with possibly discontinuous payoff functions. In this paper, we prove that the quasiconcavity assumption in Reny’s theorem can be weakened: we introduce a measure allowing to localize the lack of quasiconcavity, which allows to refine the analysis of equilibrium existence (I wish to thank P. J. Reny, two anonymous referees and the associated editor for corrections, suggestions and remarks which led to improvements in the paper).     

Existence, uniqueness and attractiveness of a pseudo almost automorphic solutions for some dissipative differential equations in Banach spaces

We give sufficient conditions ensuring the existence, uniqueness and global attractiveness of a pseudo compact almost automorphic solution of the following differential equation:

x^′(t)=f(t,x(t))     

Continuity and generators of dynamical semigroups for infinite Bose systems

For a class of quasifree quantum dynamical semigroups on the algebra of the canonical commutation relations (CCR) we give sufficient conditions for these semigroups to extend to ultraweakly continuous semigroups of normal operators on the von Neumann algebra associated with a representation of the CCR. Then the explicit form of the generators of the extended semigroups is calculated.     

Critical mass for a Patlak–Keller–Segel model with degenerate diffusion in higher dimensions

This paper is devoted to the analysis of non-negative solutions for a generalisation of the classical parabolic-elliptic Patlak–Keller–Segel system with d ≥ 3 and porous medium-like non-linear diffusion. Here, the non-linear diffusion is chosen in such a way that its scaling and the one of the Poisson term coincide. We exhibit that the qualitative behaviour of solutions is decided by the initial mass of the system. Actually, there is a sharp critical mass M _c such that if solutions exist globally in time, whereas there are blowing-up solutions otherwise. We also show the existence of self-similar solutions for . While characterising the possible infinite time blowing-up profile for M  =  M _c , we observe that the long time asymptotics are much more complicated than in the classical Patlak–Keller–Segel system in dimension two. This paper is under the Creative Commons licence Attribution-NonCommercial-ShareAlike 2.5.     

The pure displacement problem in nonlinear three-dimensional elasticity: intrinsic formulation and existence theorems

In this Note, the equations of nonlinear three-dimensional elasticity corresponding to the pure displacement problem are recast either as a boundary value problem, or as a minimization problem, where the unknown is in both cases the Cauchy–Green strain tensor, instead of the deformation as is customary. We then show that either problem possesses a solution if the applied forces are sufficiently small and the stored energy function satisfies specific hypotheses. The second problem provides an example of a minimization problem for a non-coercive functional over a Banach manifold. To cite this article: P.G. Ciarlet, C. Mardare, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

On calibrated and separating sub-actions

We consider a one-sided transitive subshift of finite type σ: Σ → Σ and a Hölder observable A. In the ergodic optimization model, one is interested in properties of A-minimizing probability measures. If ā denotes the minimizing ergodic value of A, a sub-action u for A is by definition a continuous function such that A ≥ u ○ σ ? u + ā. We call contact locus of u with respect to A the subset of Σ where A = u ○ σ ? u + ā. A calibrated sub-action u gives the possibility to construct, for any point x ε Σ, backward orbits in the contact locus of u. In the opposite direction, a separating sub-action gives the smallest contact locus of A, that we call Ω(A), the set of non-wandering points with respect to A.We prove that separating sub-actions are generic among Hölder sub-actions. We also prove that, under certain conditions on Ω(A), any calibrated sub-action is of the form u(x) = u(x _i ) + h _A (x _i , x) for some x _i ∈ Ω(A), where h _A (x, y) denotes the Peierls barrier of A. We present the proofs in the holonomic optimization model, a formalism which allows to take into account a two-sided transitive subshift of finite type \((\hat \Sigma , \hat \sigma )\).     

Theoretical investigation on 1H and 13C NMR chemical shifts of small alkanes and chloroalkanes

Using density functional theory (DFT) with the B3LYP, PBE, and PBE0 exchange-correlation functionals as well as the Moller-Plesset second-order perturbation theory (MP2) combined with a series of rather extended basis sets, 1H and 13C chemical shifts of small alkanes and chloroalkanes (with different numbers of chlorine atoms on specific positions) have been simulated and compared to experimental data. For the 1H chemical shifts, theory tends to reproduce experiment within the limits of the experimental errors. In the case of 13C chemical shift, the differences between theory and experiment increase monotonically with the number of chlorine atoms and exhibit a deviation from additivity. This behavior is related to the saturation of the experimental 13C chemical shifts with the number of chlorine atoms, whereas the evolution is mostly linear at both DFT and MP2 levels of approximation. This difference has been traced back to the relativistic spin-orbit coupling effects, which are exalted as a result of the enhancement of the s character of the C atom when increasing the number of linked Cl atoms. Thus, it was demonstrated that not only electron correlation but also relativistic effects have to be considered for estimating the 13C chemical shifts when several Cl atoms are directly attached to the C atom. Linear (theory/experiment) regressions have then been performed for the different types of C atoms, i.e., bearing one, two, and three Cl atoms, with excellent correlation coefficients. The linear correlation relationships so obtained can then serve to predict and facilitate the interpretation of the nuclear magnetic resonance spectra of more complex compounds. Furthermore, by investigating the basis set effects, the correlation between the chemical shifts calculated using the 6-311 + G(2d,p) basis set and the more extended 6-311 + G(2df,p) and aug-cc-pvtz basis sets is excellent, demonstrating that the choice of the 6-311 + G(2d,p) basis set for calculating the 1H and 13C chemical shifts is relevant.     

Chemo-regioselectivity in heterogeneous catalysis: competitive routes for C = O and C = C hydrogenations from a theoretical approach

The usual empirical rule stating that the C=C bond is more reactive than the C=O group for catalytic hydrogenations of unsaturated aldehydes is invalidated from the present study. Density functional theory calculations of all the competitive hydrogenation routes of acrolein on Pt(111) reveals conversely that the attack at the C=O bond is systematically favored. The explanation of such catalytic behavior is the existence of metastable precursor states for the O-H bond formation showing that the attack at the oxygen atom follows a new preferential mechanism where the C=O moiety is not directly bonded with the Pt surface atoms, hence yielding an intermediate pathway between Langmuir-Hinshelwood and Rideal-Eley general types of mechanisms. When the whole catalytic cycle is considered, our results reconcile with experimental studies devoted to hydrogenation of acrolein on Pt, since the desorption step of the partially hydrogenated product (unsaturated alcohol versus saturated aldehyde) plays a key role for the selectivity.