Elliptic problems with discontinuities

In this paper we prove two existence theorems for elliptic problems with discontinuities. The first one is a noncoercive Dirichlet problem and the second one is a Neumann problem. We do not use the method of upper and lower solutions. For Neumann problems we assume that f is nondecreasing. We use the critical point theory for locally Lipschitz functionals.     

Optimal control and admissible relaxation for subdifferential evolution inclusions

In this paper we study the relaxation of optimal control problems monitored by subdifferential evolution inclusions. First under appropriate convexity conditions, we establish an existence result. Then we introduce the relaxed problem and show that it always has a solution under fairly general hypotheses on the data. Subsequently we examine when the relaxation is admissible. So we show that every relaxed trajectory can be approximated by extremal original ones (i.e. original trajectories generated by bang-bang controls) and that the values of the original and relaxed problems are equal. Some examples are also presented.     

Embeddings of 2-spheres in 4-manifolds

We prove that the action of the orthogonal group of an integral, unimodular lattice whose rank and signature satisfy a certain indefiniteness condition acts transitively on pairs of mutually orthogonal elements with the same isometric invariants.     

The swelling interface number as a criterion for prediction of diffusional solute release mechanisms in swellable polymers

When a glassy polymer containing a uniformly dispersed solute is brought in contact with a penetrant, solute diffusion will be associated with the transport mechanism and penetration velocity of the penetrant in the polymer. Analysis and prediction of mechanisms of diffusional solute release may be obtained through a new dimensionless number, the swelling interface number, Sw, which compares the relative mobilities of the penetrant and the solute in the presence of macromolecular relaxations in the polymer. It is shown that a sufficient and necessary criterion for time-independent diffusional solute release rates from these swellable systems is that the Sw be smaller than 10^?2. The swelling interface number Sw may be related to easily determined structural and thermodynamic parameters of the solute/polymer/penetrant system. Preliminary experimental results of dynamic water swelling of poly(2-hydroxyethyl methacrylate-co-methyl methacrylate) and diffusional release of theophylline from initially glassy copolymers show that decreasing values of Sw are related to increased pseudo-case-II transport kinetics of the solute.     

Modelling of drug diffusion through swellable polymeric systems

A drug diffusion model for the case of diffusion of an initially uniformly distributed drug through a polymeric matrix is presented and solved. Drug diffusion from a single surface is analyzed for the case of countercurrent diffusion of a solvent which is thermodynamically compatible with the polymer. Due to swelling, considerable volume expansion is observed leading to a moving-boundary diffusion problem. Drug concentration profiles within the polymer and drug release rates can be determined. The results are in agreement with experimental data obtained for the system of KCl distributed in hydrophilic hydroxypropyl methyl cellulose matrices, in the form of tablets.     

On logarithmic Sobolev inequalities for higher order fractional derivatives

On ${\mathbb{R}}^n$, we prove the existence of sharp logarithmic Sobolev inequalities with higher fractional derivatives. Let s be a positive real number. Any function f ∈ $H^s({\mathbb{R}}^n)$ satisfies

$\underset{{\mathbb{R}}^n}{\int }|f(x){|}^2\ln (\frac{{|f(x)|}^2}{6f{6}_2^2})\mathrm{d}x+(n+\frac{n}{s}\ln \alpha +\ln \frac{s\mathrm{\Gamma }(\frac{n}{2})}{\mathrm{\Gamma }(\frac{n}{2s})})6f{6}_2^2?\frac{{\alpha }^2}{{\pi }^s}6{(?\mathrm{\Delta })}^{s/2}f{6}_2^2$

with $\alpha >0$ be any number and where the operators ${(?\mathrm{\Delta })}^{s/2}$ in Fourier spaces are defined by $\stackrel{?}{{(?\mathrm{\Delta })}^{s/2}f}(k)\text{:}={(2\pi |k|)}^s\stackrel{?}{f}(k)$. To cite this article: A. Cotsiolis, N.K. Tavoularis, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

The string coupling accelerates the expansion of the universed

Generic cosmological models in non-critical string theory have a time-dependent dilaton background at a late epoch. The cosmological deceleration parameter q ₀ is given by the square of the string coupling, g _s ², up to a negative sign. Hence the expansion of the Universe must accelerate eventually, and the observed value of q ₀ corresponds to g _s ² ∼ 0.6. In this scenario, the string coupling is asymptotically free at large times, but its present rate of change is imperceptibly small. First Award in the 2005 Essay Competition of the Gravity Research Foundation. - Ed.     

Existence of Solutions and of Multiple Solutions for Nonlinear Nonsmooth Periodic Systems

In this paper we examine nonlinear periodic systems driven by the vectorial p-Laplacian and with a nondifferentiable, locally Lipschitz nonlinearity. Our approach is based on the nonsmooth critical point theory and uses the subdifferential theory for locally Lipschitz functions. We prove existence and multiplicity results for the sublinear problem. For the semilinear problem (i.e. p = 2) using a nonsmooth multidimensional version of the Ambrosetti-Rabinowitz condition, we prove an existence theorem for the superlinear problem. Our work generalizes some recent results of Tang (PAMS 126(1998)).     

Two nontrivial solutions for quasilinear periodic equations

In this paper we study a nonlinear periodic problem driven by the ordinary scalar p-Laplacian and with a Carathéodory nonlinearity. We establish the existence of at least two nontrivial solutions. Our approach is variational based on the smooth critical point theory and using the ``Second Deformation Theorem".

     

Anisotropic diffraction and elliptic discrete solitons in two-dimensional waveguide arrays

We demonstrate that both the linear (diffraction) and the nonlinear dynamics of two-dimensional waveguide arrays are considerably more complex and versatile than their one-dimensional counterparts. The discrete diffraction properties of these arrays can be effectively altered, depending on the propagation Bloch k-vector within the first Brillouin zone of the lattice. In general, this diffraction behavior is anisotropic and therefore permits the existence of a new class of discrete elliptic solitons in the nonlinear regime.