Classification of anti-symmetric solutions to the fractional Lane-Emden system

We study the anti-symmetric solutions to the Lane-Emden type system involving fractional Laplacian(-?)^s(0 < s < 1). First, we obtain a Liouville type theorem in the often-used defining space L_2s.An interesting lower bound on the solutions is derived to estimate the Lipschitz coefficient in the sub-linear cases. Considering the anti-symmetric property, one can naturally extend the defining space from L₂ s to L_2s+1.Surprisingly, with this extension...     

Regularity of solutions for an integral system of Wolff type

We consider the fully nonlinear integral systems involving Wolff potentials:(1) whereThis system includes many known systems as special cases, in particular, when and γ=2, system (1) reduces to(2) The solutions (u,v) of (2) are critical points of the functional associated with the well-known Hardy–Littlewood–Sobolev inequality. We can show that (2) is equivalent to a system of semi-linear elliptic PDEs which comprises the well-known Lane–Emden system and Yamabe equation.We obtain integrability and regularity for the positive solutions to systems (1). A regularity lifting method by contracting operators is used in proving the integrability, and while deriving the Lipschitz continuity, a brand new idea – Lifting Regularity by Shrinking Operators is introduced. We hope to see many more applications of this new idea in lifting regularities of solutions for nonlinear problems.     

Formation and characteristics of starch‐based dual photo‐function composite hydrogel

Starch‐based dual photo‐functional hydrogel (SDPH) was obtained via incorporating the azo group onto the backbone of polymeric network and titanium dioxide within the gel at the same time. The structure and composition of SDPH were confirmed with Fourier transform infrared spectrometer, ultraviolet‐visible (UV‐Vis) spectroscopy, thermogravimetric analysis, and energy scattering X‐ray spectroscopy. Scanning electron microscopy observation revealed that titanium dioxide was well dispersed within SDPH. It was found that SDPH not only showed a reversible light‐sensitive behavior but also possessed photo‐catalytic activity. Moreover, the dual photo‐characteristics could be modulated by simply varying the initial amount of titanium dioxide.     

A Maximum Entropy Method Based on Piecewise Linear Functions for the Recovery of a Stationary Density of Interval Mappings

Let S:[0,1]→[0,1] be a nonsingular transformation such that the corresponding Frobenius-Perron operator P _S :L ¹(0,1)→L ¹(0,1) has a stationary density f ^∗. We propose a maximum entropy method based on piecewise linear functions for the numerical recovery of f ^∗. An advantage of this new approximation approach over the maximum entropy method based on polynomial basis functions is that the system of nonlinear equations can be solved efficiently because when we apply Newton’s method, the Jacobian matrices are positive-definite and tri-diagonal. The numerical experiments show that the new maximum entropy method is more accurate than the Markov finite approximation method, which also uses piecewise linear functions, provided that the involved moments are known. This is supported by the convergence rate analysis of the method.     

Symmetry of solutions to some systems of integral equations

In this paper, we study some systems of integral equations, including those related to Hardy-Littlewood-Sobolev (HLS) inequalities. We prove that, under some integrability conditions, the positive regular solutions to the systems are radially symmetric and monotone about some point. In particular, we established the radial symmetry of the solutions to the Euler-Lagrange equations associated with the classical and weighted Hardy-Littlewood-Sobolev inequality.

     

On the stationary solutions of the 2D Doi-Onsager model

In this paper, we study the 2D Doi-Onsager model

(1)     

具有非局部源的退化奇异抛物方程组解的爆破

研究了一类新的包含幂函数和指数函数相耦合的具有非局部源的抛物方程组.用正则化的方法证明了局部解的存在唯一性,用上下解方法得到了整体存在和在有限时刻爆破的充分条件.     

The best constant in a weighted Hardy-Littlewood-Sobolev inequality

We prove the uniqueness for the solutions of the singular nonlinear PDE system:

(1)

In the special case when and , we classify all the solutions and thus obtain the best constant in the corresponding weighted Hardy-Littlewood-Sobolev inequality.

     

Morphological and structural evolution of mesoporous silicas in a mild buffer solution and lysozyme adsorption

Mesoporous silicas with various morphologies and structures were synthesized with the aid of 2,2,4-trimethylpentane (TMP) in the presence of nonionic surfactant P123 [(EO)20(PO)70(EO)20] as a structure-directing agent under mild reaction conditions (HAc-NaAc buffer solution, pH 4.4). The ropelike particles formed by end-to-end interconnected nanorods were obtained at a TMP/P123 weight ratio of 0.5. It is noteworthy to mention that the mesoporous nanorods have channels running parallel to the short axis. The silica hollow spheres can be obtained at a higher TMP/P123 weight ratio because of the fusion of the primary nanorods around the interface of the O/W emulsion. Initial synthesis temperatures of 15, 25, and 40 degrees C can lead to mesoporous silicas with highly ordered 2D hexagonal mesostructure, vesicular mesostructure, and mesostructured cellular foams (MCF), respectively. The mesoporous silicas exhibit high adsorption capacity (up to 536 mg g(-1)) and very rapid (<5 min to reach equilibrium) lysozyme immobilization. More importantly, it is revealed that mesoporous silica hollow spheres with rugged surfaces can greatly accelerate the adsorption rate of the enzyme during the adsorption process.