Excellent UV-absorbing cotton fabric with high durability and hydrophobicity based on surface-initiated ATRP of polymerizable organic UV-absorber

Cellulose - Ultraviolet (UV) protective cotton fabric is a convenient and reliable way to protect human body from sunlight. Herein, a polymerizable UV-absorber,...     

Radial symmetry and regularity of solutions for poly-harmonic Dirichlet problems

Let B=B₁(0) be the unit ball in Rn and r=|x|. We study the poly-harmonic Dirichlet problem

     

Hardy–Littlewood–Sobolev and Stein–Weiss inequalities and integral systems on the Heisenberg group

In this paper, we study two types of weighted Hardy–Littlewood–Sobolev (HLS) inequalities, also known as Stein–Weiss inequalities, on the Heisenberg group. More precisely, we prove the |u|$\left|u\right|$ weighted HLS inequality in Theorem 1.1 and the |z|$\left|z\right|$ weighted HLS inequality in Theorem 1.5 (where we have denoted u=(z,t)$u=(z,t)$ as points on the Heisenberg group). Then we provide regularity estimates of positive solutions to integral systems which are Euler–Lagrange equations of the possible extremals to the Stein–Weiss inequalities. Asymptotic behavior is also established for integral systems associated to the |u|$\left|u\right|$ weighted HLS inequalities around the origin. By these a priori estimates, we describe asymptotically the possible optimizers for sharp versions of these inequalities.     

Liouville-type theorems and decay estimates for solutions to higher order elliptic equations

Liouville-type theorems are powerful tools in partial differential equations. Boundedness assumptions of solutions are often imposed in deriving such Liouville-type theorems. In this paper, we establish some Liouville-type theorems without the boundedness assumption of nonnegative solutions to certain classes of elliptic equations and systems. Using a rescaling technique and doubling lemma developed recently in Polá?ik et al. (2007) [20], we improve several Liouville-type theorems in higher order elliptic equations, some semilinear equations and elliptic systems. More specifically, we remove the boundedness assumption of the solutions which is required in the proofs of the corresponding Liouville-type theorems in the recent literature. Moreover, we also investigate the singularity and decay estimates of higher order elliptic equations.     

Symmetry and regularity of extremals of an integral equation related to the Hardy�CSobolev inequality

Let ?? be a real number satisfying 0?<????<?n, ${0\leq t<\alpha, \alpha{^\ast}(t)=\frac{2(n-t)}{n-\alpha}}$ . We consider the integral equation $$u(x)=\int\limits_{{\mathbb{R}^n}}\frac{u^{{\alpha{^\ast}(t)}-1}(y)}{|y|^t|x-y|^{n-\alpha}}\,dy,\quad\quad\quad\quad\quad\quad\quad(1)$$ which is closely related to the Hardy?CSobolev inequality. In this paper, we prove that every positive solution u(x) is radially symmetric and strictly decreasing about the origin by the method of moving plane in integral forms. Moreover, we obtain the regularity of solutions to the following integral equation $$u(x)=\int\limits_{{\mathbb{R}^n}}\frac{|u(y)|^{p}u(y)}{|y|^t|x-y|^{n-\alpha}}\, dy\quad\quad\quad\quad\quad\quad\quad(2)$$ that corresponds to a large class of PDEs by regularity lifting method.     

Analysis of the contribution of sedimentation to bacterial mass transport in a parallel plate flow chamber

In order to investigate bacterium-substratum interactions, understanding of bacterial mass transport is necessary. Comparisons of experimentally observed initial deposition rates with mass transport rates in parallel-plate-flow-chambers (PPFC) predicted by convective-diffusion yielded deposition efficiencies above unity, despite electrostatic repulsion. It is hypothesized that sedimentation is the major mass transport mechanism in a PPFC. The contribution of sedimentation to the mass transport in a PPFC was experimentally investigated by introducing a novel microscopy-based method. First, height-dependent bacterial concentrations were measured at different times and flow rates and used to calculate bacterial sedimentation velocities. For Staphylococcus aureus ATCC 12600, a sedimentation velocity of 240 μm h(-1) was obtained. Therewith, sedimentation appeared as the predominant contribution to mass transport in a PPFC. Also in the current study, deposition efficiencies of S. aureus ATCC 12600 with respect to the Smoluchowski-Levich solution of the convective-diffusion equation were four-to-five fold higher than unity. However, calculation of deposition efficiencies with respect to sedimentation were below unity and decreased from 0.78 to 0.36 when flow rates increased from 0.017 to 0.33 cm(3) s(-1). The proposed analysis of bacterial mass transport processes is simple, does not require additional equipment and yields a more reasonable interpretation of bacterial deposition in a PPFC.     

Model smoke stream adsorption over cellulose acetate stick with three-dimensional temperature gradient by combining in-situ DRIFTS with infrared thermal imaging

Cellulose - Understanding the adsorption of the smoke stream (SR) on cigarette filter—cellulose acetate stick with different temperatures is beneficial for controlling chemical emissions and...     

A priori estimates, existence and non-existence of positive solutions of generalized mean curvature equations

This paper concerns a priori estimates and existence of solutions of generalized mean curvature equations with Dirichlet boundary value conditions in smooth domains. Using the blow-up method with the Liouville-type theorem of the p laplacian equation, we obtain a priori bounds and the estimates of interior gradient for all solutions. The existence of positive solutions is derived by the topological method. We also consider the non-existence of solutions by Pohozaev identities.