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 $d^{⁎}\in \{5,6,7\}$.     

Nickel tetrathiooxalate as a cathode material for potassium batteries

We report a nickel tetrathiooxalate (NiTTO) coordination polymer as a cathode material for potassium batteries. In a potential range of 1.3–3.6 V vs. K⁺/K, the specific capacity of the material is 209 mA h g^?1 at a current density of 0.1 A g^?1, which roughly corresponds to the two-electron reduction of polymer repeating units. The charge–discharge mechanisms of NiTTO in potassium cells were examined using operando Raman spectroscopy.     

Design of Functional Carbon Composite Materials for Energy Conversion and Storage

The carbon composite materials have been a research hotspot in the fields of catalysis, energy conversion and so on, because of their features of large structure and morphology variety, good chemical and electrochemical stability, and high electronic conductivity, large specific surface area and rich active sites. This paper summarizes some research progress of carbon composite materials, including assembly methodologies, their structure regulation, properties, and related applications. Moreover, the current challenges and the prospects of these materials are also discussed.     

Visible-Light Augmented Lithium Storage Capacity in a Ruthenium(II) Photosensitizer Conjugated with a Dione-Catechol Redox Couple

Controlling redox activity of judiciously appended redox units on a photo-sensitive molecular core is an effective strategy for visible light energy harvesting and storage. The first example of a photosensitizer - electron donor coordination compound in which the photoinduced electron transfer step is used for light to electrical energy conversion and storage is reported. A photo-responsive Ru-diimine module conjugated with redox-active catechol groups in [Ru(II)(phenanthroline-5,6-diolate)₃]⁴⁻ photosensitizer can mediate photoinduced catechol to dione oxidation in the presence of a sacrificial electron acceptor or at the surface of an electrode. Under potentiostatic condition, visible light triggered current density enhancement confirmed the light harvesting ability of this photosensitizer. Upon implementation in galvanostatic charge-discharge of a Li battery configuration, the storage capacity was found to be increased by 100 %, under 470 nm illumination with output power of 4.0 mW/cm⁻². This proof-of-concept molecular system marks an important milestone towards a new generation of molecular photo-rechargeable materials.     

Decay rate for viscoelastic wave equation of variable coefficients with delay and dynamic boundary conditions

In this paper, we consider a viscoelastic wave equation of variable coefficients in the presence of past history with nonlinear damping and delay in the internal feedback and dynamic boundary conditions. Under suitable assumptions, we establish an explicit and general decay rate result without imposing restrictive assumption on the behavior of the relaxation function at infinity by Riemannian geometry method and Lyapunov functional method.     

Micro/Nanoengineered α-Fe2O3 Nanoaggregate Conformably Enclosed by Ultrathin N-Doped Carbon Shell for Ultrastable Lithium Storage and Insight into Phase Evolution Mechanism

The Fe-based transition metal oxides are promising anode candidates for lithium storage considering their high specific capacity, low cost, and environmental compatibility. However, the poor electron/ion conductivity and significant volume stress limit their cycle and rate performances. Furthermore, the phenomena of capacity rise and sudden decay for α-Fe₂O₃ have appeared in most reports. Here, a uniform micro/nano α-Fe₂O₃ nanoaggregate conformably enclosed in an ultrathin N-doped carbon network (denoted as M/N-α-Fe₂O₃@NC) is designed. The M/N porous balls combine the merits of secondary nanoparticles to shorten the Li⁺ transportation pathways as well as alleviating volume expansion, and primary microballs to stabilize the electrode/electrolyte interface. Furthermore, the ultrathin carbon shell favors fast electron transfer and protects the electrode from electrolyte corrosion. Therefore, the M/N-α-Fe₂O₃@NC electrode delivers an excellent reversible capacity of 901 mA h g⁻¹ with capacity retention up to 94.0 % after 200 cycles at 0.2 A g⁻¹. Notably, the capacity rise does not happen during cycling. Moreover, the lithium storage mechanism is elucidated by ex situ XRD and HRTEM experiments. It is verified that the reversible phase transformation of α↔γ occurs during the first cycle, whereas only the α-Fe₂O₃ phase is reversibly transformed during subsequent cycles. This study offers a simple and scalable strategy for the practical application of high-performance Fe₂O₃ electrodes.     

Analysis of a Cahn–Hilliard–Brinkman model for tumour growth with chemotaxis

Phase field models recently gained a lot of interest in the context of tumour growth models. Typically Darcy-type flow models are coupled to Cahn–Hilliard equations. However, often Stokes or Brinkman flows are more appropriate flow models. We introduce and mathematically analyse a new Cahn–Hilliard–Brinkman model for tumour growth allowing for chemotaxis. Outflow boundary conditions are considered in order not to influence tumour growth by artificial boundary conditions. Existence of global-in-time weak solutions is shown in a very general setting.