Regularity of optimal sets for some functional involving eigenvalues of an operator in divergence form

In this paper we consider minimizers of the functional$\min \{{\lambda }_1(\mathrm{\Omega })+\cdots +{\lambda }_k(\mathrm{\Omega })+\mathrm{\Lambda }|\mathrm{\Omega }|,:\mathrm{\Omega }\subset D\text{ open}\}$ where $D\subset {\mathbb{R}}^d$ is a bounded open set and where $0<{\lambda }_1(\mathrm{\Omega })\le \cdots \le {\lambda }_k(\mathrm{\Omega })$ are the first k eigenvalues on Ω of an operator in divergence form with Dirichlet boundary condition and with Hölder continuous coefficients. We prove that the optimal sets ${\mathrm{\Omega }}^{⁎}$ have finite perimeter and that their free boundary $\partial {\mathrm{\Omega }}^{⁎}\cap D$ is composed of a regular part, which is locally the graph of a $C^{1,\alpha }$-regular function, and a singular part, which is empty if $d<d^{⁎}$, discrete if $d=d^{⁎}$ and of Hausdorff dimension at most $d-d^{⁎}$ if $d>d^{⁎}$, for some $d^{⁎}\in \{5,6,7\}$.     

Developing Flexible Quinacridone-Derivatives-Based Photothermal Evaporaters for Solar Steam and Thermoelectric Power Generation

Solar-driven interfacial vaporization by localizing solar-thermal energy conversion to the air−water interface has attracted tremendous attention. In the process of converting solar energy into heat energy, photothermal materials play an essential role. Herein, a flexible solar-thermal material di-cyan substituted 5,12-dibutylquinacridone (DCN−4CQA)@Paper was developed by coating photothermal quinacridone derivatives on the cellulose paper. The DCN−4CQA@Paper combines desired chemical and physical properties, broadband light-absorbing, and shape-conforming abilities that render efficient photothermic vaporization. Notably, synergetic coupling of solar-steam and solar-electricity technologies by integrating DCN−4CQA@Paper and the thermoelectric devices is realized without trade-offs, highlighting the practical consideration toward more impactful solar heat exploitation. Such solar distillation and low-grade heat-to-electricity generation functions can provide potential opportunities for fresh water and electricity supply in off-grid or remote areas.     

Statistical Optimization of Flavonoid and Antioxidant Recovery from Macerated Chinese and Malaysian Lotus Root (Nelumbo nucifera) Using Response Surface Methodology

In this study, the combination of parameters required for optimal extraction of anti-oxidative components from the Chinese lotus (CLR) and Malaysian lotus (MLR) roots were carefully investigated. Box–Behnken design was employed to optimize the pH (X₁: 2–3), extraction time (X₂: 0.5–1.5 h) and solvent-to-sample ratio (X₃: 20–40 mL/g) to obtain a high flavonoid yield with high % DPPH_sc free radical scavenging and Ferric-reducing power assay (FRAP). The analysis of variance clearly showed the significant contribution of quadratic model for all responses. The optimal conditions for both Chinese lotus (CLR) and Malaysian lotus (MLR) roots were obtained as: CLR: X₁ = 2.5; X₂ = 0.5 h; X₃ = 40 mL/g; MLR: X₁ = 2.4; X₂ = 0.5 h; X₃ = 40 mL/g. These optimum conditions gave (a) Total flavonoid content (TFC) of 0.599 mg PCE/g sample and 0.549 mg PCE/g sample, respectively; (b) % DPPH_sc of 48.36% and 29.11%, respectively; (c) FRAP value of 2.07 mM FeSO₄ and 1.89 mM FeSO₄, respectively. A close agreement between predicted and experimental values was found. The result obtained succinctly revealed that the Chinese lotus exhibited higher antioxidant and total flavonoid content when compared with the Malaysia lotus root at optimum extraction condition.     

Semilinear wave models with friction and viscoelastic damping

In this paper, we study the global (in time) existence of small data solutions to the Cauchy problem for the semilinear wave equation with friction, viscoelastic damping, and a power nonlinearity. We are interested in the connection between regularity assumptions for the data and the admissible range of exponents p in the power nonlinearity.     

Analysis on X-Ray Laser Generation with EM-Wave Wiggler

The possibility of realizing SASE Free Election Laser in x-ray waveband with EM-wave wiggler is studied in this paper. SASE effect including saturation length, saturation power and nonlinear process is explored through a 3-D simulation code. A utilization of seed light from laser plasma x-ray is also analyzed, which demonstrates the feature of shortening the interaction length for saturation. The results show that sizeable output power of x-ray laser would be generated with a middle energy electron beam.     

Boundedness of commutators on homogeneous Herz spaces

The boundedness on homogeneous Herz spaces is established for a large class of linear commutators generated by BMO(R ⁿ ) functions and linear operators of rough kernels which include the Calderón-Zygmund operators and the Ricci-Stein oRfiUatory singular integrals with rough kernels. Project supponed in pan by the National h’atural Science Foundation of China (Grant No. 19131080) and the NEDF of China.     

Equivalence of variational inequality problems to unconstrained minimization

In this paper we propose a class of merit functions for variational inequality problems (VI). Through these merit functions, the variational inequality problem is cast as unconstrained minimization problem. We estimate the growth rate of these merit functions and give conditions under which the stationary points of these functions are the solutions of VI. This work was supported by the state key project “Scientific and Engineering Computing”.     

环状连续强激光下光学薄膜的瞬温和热畸变

详细研究了非冷却光学薄膜元件在环形激光光束辐照下的瞬态温度分布和随后的热畸变，并且用泰曼干涉仪实际测量了连续波氧碘激光辐照各种非冷却光学元件的热畸变量并与理论计算相比较，分析结果表明理论和实验相一致。