The Neumann Problem and Helmholtz Decomposition in Convex Domains

We show that the Neumann problem for Laplace's equation in a convex domain $\Omega$ with boundary data in $L^p(\partial\Omega)$ is uniquely solvable for $1<p<\infty$. As a consequence, we obtain the Helmholtz decomposition of vector fields in $L^p(\Omega, \mathbb{R}^d)$.Comment: 17 page

Uniform W^{1,p} Estimates for Systems of Linear Elasticity in a Periodic Medium

Let $\mathcal{L}_\epsilon$ be a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients. We obtain the uniform $W^{1,p}$ estimate in a Lipschitz domain for solutions to the Dirichlet problem, where $(2n/(n+1)) -\delta<p<(2n/(n-1))+\delta$. The ranges of $p$'s are sharp for $n=2$ or 3

INVESTIGATION OF THE TOXICITY AND EFFLUX OF POLYCHLORINATED BIPHENYLS AND HYDROXYLATED POLYCHLORINATED BIPHENYLS IN \u3cem\u3eESCHERICHIA COLI\u3c/em\u3e

Polychlorinated biphenyls (PCBs) are persistent organic pollutants. Due to their properties, PCBs accumulate in the food-chain and post a threat to the health of human beings and wildlife. Hydroxylated PCBs (OH-PCBs) are oxidative metabolites of PCBs and are more hydrophilic than their parent PCBs. One of the best approaches to break down these contaminants is through bioremediation, which is an environmental friendly process that uses microorganisms to restore natural environment. Towards this goal, we have investigated the toxicity and accumulation of PCBs and OH-PCBs in a Gram-negative bacterium, Escherichia coli. We have also determined the role played by a primary multidrug efflux transporter AcrB on the accumulation of PCBs and OH-PCBs in bacterial cell. We found that one of the PCBs tested was toxic to E. coli, while different OH-PCBs have different levels of toxicity; the acrB knockout strain accumulated significantly more PCBs and OH-PCBs than the wild-type strain, suggesting that these compounds are substrates of the efflux pump; higher cytoplasmic concentrations of OH-PCBs were also observed in the acrB knockout strain using the biosensors. Based on these observations, we conclude that both PCBs and OH-PCBs are substrates of protein AcrB. Therefore the efflux activities of multidrug resistant pumps in Gram-negative bacteria should be considered while designing bioremediation approaches

Matter Power Spectra in Viable f(R)f(R) Gravity Models with Massive Neutrinos

We investigate the matter power spectra in the power law and exponential types of viable $f(R)$ theories along with massive neutrinos. The enhancement of the matter power spectrum is found to be a generic feature in these models. In particular, we show that in the former type, such as the Starobinsky model, the spectrum is magnified much larger than the latter one, such as the exponential model. A greater scale of the total neutrino mass, $\Sigma m_{\nu}$, is allowed in the viable $f(R)$ models than that in the $\Lambda$CDM one. We obtain the constraints on the neutrino masses by using the CosmoMC package with the modified MGCAMB. Explicitly, we get $\Sigma m_{\nu} < 0.451 \ (0.214)\ \mathrm{eV}$at 95the corresponding one for the$\Lambda$CDM model is$\Sigma m_{\nu} < 0.200\ \mathrm{eV}$. Furthermore, by treating the effective number of neutrino species$N_{\mathrm{eff}}$as a free parameter along with$\Sigma m_{\nu}$, we find that$N_{\mathrm{eff}} = 3.78^{+0.64}_{-0.84} (3.47^{+0.74}_{-0.60})$and$\Sigma m_{\nu} = 0.533^{+0.254}_{-0.411}$($< 0.386) \ \mathrm{eV}$ at 95% C.L. in the Starobinsky (exponential) model.Comment: 15 pages, 5 figures, updated version accepted by PL