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.     

Nonlinear obstacle problems with double phase in the borderline case

In this paper we study a double phase problem with an irregular obstacle. The energy functional under consideration is characterized by the fact that both ellipticity and growth switch between a type of polynomial and a type of logarithm, which can be regarded as a borderline case of the double phase functional with $(p,q)$-growth. We obtain an optimal global Calderón–Zygmund type estimate for the obstacle problem with double phase in the borderline case.     

PVT法氮化铝晶体铝面、氮面生长对比分析

在氮气环境下用PVT方法生长氮化铝过程中,氮面和铝面由于表面化学性质不同,生长的主要化学反应速度存在差异。原子在生长表面的迁移能力不同造成单晶表面生长方式差异较大。在基本相同条件下(生长温度、生长温差、生长气压、类似的籽晶、同一台生长设备)进行了铝、氮面氮化铝单晶晶体生长。为了更明显地表现铝氮面的差异,将同一片籽晶分为两半,翻转其中一半让铝氮面同时生长。铝面生长较好的区域形成了明显的晶畴,而氮面生长时生长较好的部分出现了明显的生长台阶,并出现了晶畴边界被生长台阶湮灭的生长现象,进一步通过AFM观测到铝面生长台阶平整但被缺陷所阻隔,晶畴发育明显为各晶畴独立生长。氮面生长台阶没有铝面规则但连续性较强,在原来晶畴边界位置也出现了连续的生长台阶(或台阶簇)。所以籽晶质量不高时氮面生长更容易提高晶体质量,后续的XRD测量结果也证明了氮面生长后的晶体质量明显高于铝面生长的晶体质量。     

Curved domain walls dynamics driven by magnetic field and electric current in hard ferromagnets

The propagation of curved domain walls in hard ferromagnetic materials is studied by applying a reductive perturbation method to the generalized Landau–Lifshitz–Gilbert equation. The extended model herein considered explicitly takes into account the effects of a spin-polarized current as well as those arising from a nonlinear dissipation.     

On analyticity of the ‐Stokes semigroup for some non‐Helmholtz domains

Consider the Stokes equations in a sector‐like C ³ domain . It is shown that the Stokes operator generates an analytic semigroup in for . This includes domains where the ‐Helmholtz decomposition fails to hold. To show our result we interpolate results of the Stokes semigroup in and L ² by constructing a suitable non‐Helmholtz projection to solenoidal spaces.     

Composite generalized Laguerre spectral method for nonlinear Fokker–Planck equations on the whole line

In this paper, we propose a composite Laguerre spectral method for the nonlinear Fokker–Planck equations modelling the relaxation of fermion and boson gases. A composite Laguerre spectral scheme is constructed. Its convergence is proved. Numerical results show the efficiency of this approach and coincide well with theoretical analysis. Some results on the Laguerre approximation and techniques used in this paper are also applicable to other nonlinear problems on the whole line. Copyright © 2016 John Wiley & Sons, Ltd.     

Optimal control problem in a domain with branched structure and homogenization

We consider a linear parabolic problem in a thick junction domain which is the union of a fixed domain and a collection of periodic branched trees of height of order 1 and small width connected on a part of the boundary. We consider a three‐branched structure, but the analysis can be extended to n‐branched structures. We use unfolding operator to study the asymptotic behavior of the solution of the problem. In the limit problem, we get a multi‐sheeted function in which each sheet is the limit of restriction of the solution to various branches of the domain. Homogenization of an optimal control problem posed on the above setting is also investigated. One of the novelty of the paper is the characterization of the optimal control via the appropriately defined unfolding operators. Finally, we obtain the limit of the optimal control problem. Copyright © 2016 John Wiley & Sons, Ltd.