Solvability of the Dirichlet, Neumann and the regularity problems for parabolic equations with H\"older continuous coefficients

We establish the $L^2$-solvability of Dirichlet, Neumann and regularity problems for divergence-form heat (or diffusion) equations with H\"older-continuous diffusion coefficients, on bounded Lipschitz domains in $\mathbb{R}^n$. This is achieved through the demonstration of invertibility of the relevant layer-potentials which is in turn based on Fredholm theory and a new systematic approach which yields suitable parabolic Rellich-type estimates

Regional income convergence and regional policy in the European Union

In this paper we use a generalized entropy index such as the Theil index to analyze regional inequalities in Europe. We proved that there is a synchronization between the convergence and catching-up process of objective 1 regions towards the EU15 average with the reform of the EU regional policy. During the period 1982-1988 the Theil index shows that inequalities between objective 1 regions and non-objective 1 regions have increased while from 1989 onwards the reduction in the inequalities between these two groups has been the norm. We also remark the fact that there are high disparate rates of growth among objective 1 regions both within countries and across countries but our computations show also a trend towards a more balanced growth among objective 1 regions within and across EU countries. This success of the European Union regional policy in objective 1 regions will mean a big opportunity for Central and Eastern European countries and hence the increases in competition arising from an enlarged European market combined with a suitable regional development policy should in the future boost the growth of those countries. In the last part of the paper we made a simulation for the funding envelope from 2007, based on the 2000-2006 budget. We show that the figures of the Agenda 2000 provide enough financial support for 90% of the total CEEC population and for 75% of “current” objective 1 population. Key Words: Regional Policy, European Enlargement, Central and Eastern European Countries, Strategic Planning, Regional Growth, Regional Development

Transference of local to global L2L^2 maximal estimates for dispersive partial differential equations

In this paper we give an elementary proof for transference of local to global maximal estimates for dispersive PDEs. This is done by transferring local $L^2$ estimates for certain oscillatory integrals with rough phase functions, to the corresponding global estimates. The elementary feature of our approach is that it entirely avoids the use of the wave packet techniques which are quite common in this context, and instead is based on scalings and classical oscillatory integral estimates.Comment: 10 page

Large amplitude pairing fluctuations in atomic nuclei

Pairing fluctuations are self-consistently incorporated on the same footing as the quadrupole deformations in present state of the art calculations including particle number and angular momentum conservation as well as configuration mixing. The approach is complemented by the use of the finite range density dependent Gogny force which, with a unique source for the particle-hole and particle-particle interactions, guarantees a self-consistent interplay in both channels. We have applied our formalism to study the role of the pairing degree of freedom in the description of the most relevant observables like spectra, transition probabilities, separation energies, etc. We find that the inclusion of pairing fluctuations mostly affects the description of excited states, depending on the excitation energy and the angular momentum. $E0$ transition probabilities experiment rather big changes while $E2$'s are less affected. Genuine pairing vibrations are thoroughly studied with the conclusion that deformations strongly inhibits their existence. These studies have been performed for a selection of nuclei: spherical, deformed and with different degree of collectivity.Comment: 23 pages, 23 Figures, To be published in Phys. Rev.

Study of sdO models. Pulsation Analysis

We have explored the possibility of driving pulsation modes in models of sdO stars in which the effects of element diffusion, gravitational settling and radiative levitation have been neglected so that the distribution of iron-peak elements remains uniform throughout the evolution. The stability of these models was determined using a non-adiabatic oscillations code. We analysed 27 sdO models from 16 different evolutionary sequences and discovered the first ever sdO models capable of driving high-radial order g-modes. In one model, the driving is by a classical kappa-mechanism due to the opacity bump from iron-peak elements at temperature ~200,000 K. In a second model, the driving result from the combined action of kappa-mechanisms operating in three distinct regions of the star: (i) a carbon-oxygen partial ionization zone at temperature ~2 10^6 K, (ii) a deeper region at temperature ~2 10^7 K, which we attribute to ionization of argon, and (iii) at the transition from radiative to conductive opacity in the core of the star.Comment: 13 pages, 19 figures, accepted for publication in MNRAS, 2009 September 1

Revisiting the optical PTPT-symmetric dimer

Optics has proved a fertile ground for the experimental simulation of quantum mechanics. Most recently, optical realizations of $\mathcal{PT}$-symmetric quantum mechanics have been shown, both theoretically and experimentally, opening the door to international efforts aiming at the design of practical optical devices exploiting this symmetry. Here, we focus on the optical $\mathcal{PT}$-symmetric dimer, a two-waveguide coupler were the materials show symmetric effective gain and loss, and provide a review of the linear and nonlinear optical realizations from a symmetry based point of view. We go beyond a simple review of the literature and show that the dimer is just the smallest of a class of planar $N$-waveguide couplers that are the optical realization of Lorentz group in 2+1 dimensions. Furthermore, we provide a formulation to describe light propagation through waveguide couplers described by non-Hermitian mode coupling matrices based on a non-Hermitian generalization of Ehrenfest theorem.Comment: 25 pages, 12 figure

Study of sdO models: mode trapping

We present the first description of mode trapping for sdO models. Mode trapping of gravity modes caused by the He/H chemical transition is found for a particular model, providing a selection effect for high radial order trapped modes. Low- and intermediate-radial order {\em p}-modes (mixed modes with a majority of nodes in the P-mode region) are found to be trapped by the C-O/He transition, but with no significant effects on the driving. This region seems to have also a subtle effect on the trapping of low radial order {\em g}-modes (mixed modes with a majority of nodes in the G-mode region), but again with no effect on the driving. We found that for mode trapping to have an influence on the driving of sdO modes (1) the mode should be trapped in a way that the amplitude of the eigenfunctions is lower in a damping region and (2) in this damping region significant energy interchange has to be produced.Comment: 10 pages, 13 figures, accepted for publication in MNRAS, 2009 December 1