Weighted estimates for the multilinear maximal function on the upper half‐spaces

For a general dyadic grid, we give a Calderón–Zygmund type decomposition, which is the principle fact about the multilinear maximal function on the upper half‐spaces. Using the decomposition, we study the boundedness of . We obtain a natural extension to the multilinear setting of Muckenhoupt's weak‐type characterization. We also partially obtain characterizations of Muckenhoupt's strong‐type inequalities with one weight. Assuming the reverse Hölder's condition, we get a multilinear analogue of Sawyer's two weight theorem. Moreover, we also get Hytönen–Pérez type weighted estimates.     

Schatten properties,nuclearity and minimality of phase shift invariant spaces

We extend Feichtinger's minimality property on the smallest non-trivial time-frequency shift invariant Banach space, to the quasi-Banach case. Analogous properties are deduced for certain matrix spaces.We use these results to prove that the pseudo-differential operator $\mathrm{Op}(a)$ is a Schatten-q operator from $M^{\infty }$ to $M^p$ and r-nuclear operator from $M^{\infty }$ to $M^r$ when $a\in M^r$ for suitable p, q and r in $(0,\infty ]$.     

Phase transition layers with boundary intersection for an inhomogeneous Allen–Cahn equation

We consider the nonlinear problem of inhomogeneous Allen–Cahn equation

${?}^2\mathrm{\Delta }u+V(y)u(1?u^2)=0\text{in}\mathrm{\Omega },\frac{?u}{?\nu }=0\text{on}?\mathrm{\Omega },$

where Ω is a bounded domain in ${\mathbb{R}}^2$ with smooth boundary, ? is a small positive parameter, ν denotes the unit outward normal of ?Ω, V is a positive smooth function on $\overline{\mathrm{\Omega }}$. Let Γ be a curve intersecting orthogonally with ?Ω at exactly two points and dividing Ω into two parts. Moreover, Γ satisfies stationary and non-degenerate conditions with respect to the functional ${\int }_{\mathrm{\Gamma }}{V}^{1/2}$. We can prove that there exists a solution $u_{?}$ such that: as $?\to 0$, $u_{?}$ approaches +1 in one part of Ω, while tends to ?1 in the other part, except a small neighborhood of Γ.     

Free boundary problem for a fully nonlinear and degenerate parabolic equation in an angular domain

This paper concerns with the problem of how to running an insurance company to maximize his total discounted expected dividends. In our model, the dividend rate is limited in $[0,M]$ and the company is allowed to transfer any proportion of risk by reinsuring. So there are two strategies which we call dividend strategy and reinsurance strategy. The objective function and the corresponding optimal two strategies are the solution and the two free boundaries of the following Barenblatt parabolic equation

$v_t?\underset{0\le a\le 1}{\max }?(\frac{1}{2}{a}^2{\sigma }^2{v}_{xx}+a\mu v_x)+cv?\underset{0\le l\le M}{\max }?[(1?v_x)l]=0$

under certain boundary conditions on an angular domain

$Q_T=\{(x,t)|0<x<Mt,0<t\le T\}.$

The main effort is to analyze the properties of the solution and the free boundaries to show the optimal decision for the insurance company.     

Flower-like shaped Bi12TiO20/g-C3N4 heterojunction for effective elimination of organic pollutants: Preparation,characterization, and mechanism study

Flower-like shaped Bi₁₂TiO₂₀ (Bismuth Titanate)/g-C₃N₄ (graphite-like carbon nitride) heterojunction was prepared through hydrothermal and sonification methods for the degradation of organic pollutants by visible-light irradiation. The preparation process, chemical structures, and the mechanism of photocatalytic enhancement of the heterostructures were studied systematically. Under visible-light irradiation, the novel flower-like shaped Bi₁₂TiO₂₀/g-C₃N₄ heterojunction demonstrates prominent activities for the degradation of rhodamine B and p-nitrophenol, with the introduction of flower-like shaped Bi₁₂TiO₂₀ into g-C₃N₄ composites greatly increasing the activity of pure g-C₃N₄. This activity enhancement for the heterojunction could be mainly attributed to its low recombination speed of electron–hole pairs, high adsorption ability of organic pollutants, and better optical absorption ability. Moreover, in the visible-light system of Bi₁₂TiO₂₀/g-C₃N₄, ^•OH also contributed to the degradation of pollutants, which may explain the enhanced photocatalytic activity after the introduction of Bi₁₂TiO_20, as ^•OH is inactive in pure g-C₃N₄. Furthermore, 10 wt.% Bi₁₂TiO₂₀/g-C₃N₄ showed not only high activity but also good stability for degradation of aqueous organic pollutants, implying potential applications prospect.