Qualitative properties of singular solutions to fractional elliptic equations

In this paper, by the moving spheres method, Caffarelli-Silvestre extension formula and blow-up analysis, we study the local behaviour of nonnegative solutions to fractional elliptic equations \begin{align*} (-\Delta)^{\alpha} u =f(u),~~ x\in \Omega\backslash \Gamma, \end{align*} where $0<\alpha <1$ , $\Omega = \mathbb {R}^{N}$ or $\Omega$ is a smooth bounded domain, $\Gamma$ is a singular subset of $\Omega$ with fractional capacity zero, $f(t)$ is locally bounded and positive for $t\in [0,\,\infty )$ , and $f(t)/t^{({N+2\alpha })/({N-2\alpha })}$ is nonincreasing in $t$ for large $t$ , rather than for every $t>0$ . Our main result is that the solutions satisfy the estimate \begin{align*} f(u(x))/ u(x)\leq C d(x,\Gamma)^{{-}2\alpha}. \end{align*} This estimate is new even for $\Gamma =\{0\}$ . As applications, we derive the spherical Harnack inequality, asymptotic symmetry, cylindrical symmetry of the solutions.

Layered-rocksalt intergrown cathode for high-capacity zero-strain battery operation

The dependence on lithium-ion batteries leads to a pressing demand for advanced cathode materials. We demonstrate a new concept of layered-rocksalt intergrown structure that harnesses the combined figures of merit from each phase, including high capacity of layered and rocksalt phases, good kinetics of layered oxide and structural advantage of rocksalt. Based on this concept, lithium nickel ruthenium oxide of a main layered structure (R$$\bar{3}$$ 3 ¯ m) with intergrown rocksalt (Fm$$\bar{3}$$ 3 ¯ m) is developed, which delivers a high capacity with good rate performance. The interwoven rocksalt structure successfully prevents the anisotropic structural change that is typical for layered oxide, enabling a nearly zero-strain operation upon high-capacity cycling. Furthermore, a design principle is successfully extrapolated and experimentally verified in a series of compositions. Here, we show the success of such layered-rocksalt intergrown structure exemplifies a new battery electrode design concept and opens up a vast space of compositions to develop high-performance intergrown cathode materials.

ON HUNT PROCESSES AND STRICT CAPACITIES ASSOCIATED WITH GENERALIZED DIRICHLET FORMS

Introducing the corresponding strict capacity, we give necessary and sufficient conditions for a generalized Dirichlet form to be associated with a Hunt process. We also show that Borel measurable sets with strict capacity zero can be checked-out by an appropriate subclass of smooth measures. In the last part of this paper we present applications to three classes of examples.

Cesaro means and Dirichlet spaces of holomorphic functions

We establish convergence results involving the nth Cesaro mean σβn(θ) of order β of f(eiθ) = [sum ]aneinθ where the coefficients (an) satisfy a Dirichlet-type condition [sum ]nα[mid ]an[mid ]2 < ∞ for a fixed α in (0, 1]. When α = 1, for example, we prove that [sum ][mid ]σβn(θ)−f(eiθ) [mid ]2 < ∞ for every β > −½ at almost all points θ in [−π,π]. We also derive weighted convergence results of this type outside exceptional sets of capacity zero, and show by examples that these results are, in a certain sense, sharp with respect to the size of the exceptional sets.

Adsorption of Iodide Ion from Aqueous NaI-NaClO4- Butan-1-ol at the Mercury-Electrolyte Interface

Properties of the adsorption layers I--ClO4--H2O and I--ClO4--H2O-butan-1-ol on the mercury electrode were described using adsorption and kinetic parameters for the Zn(II) ion reduction. The systems were characterized by the measurements of differential capacity, zero charge potential and surface tension at this potential. The data were analysed to obtain the surface pressure and the charge due to specific adsorption of I- ions as a function of charge and concentration of I- ions. The standard Gibbs energy of adsorption ∆G0ad and parameter B obtained from the virial isotherm were compared. The rate constants of Zn(II) ion reduction as an ion affecting the adsorption equilibrium were determined.

The Mixed Adsorption Layers of Butanol/Thiourea and Butanol/Toluidine at the Interface Hg/Aqueous Perchlorate Solution

Properties of mixed, two-component adsorption layers of butanol/thiourea, butanol/m-toluidine and butanol/p-toluidine in 1 M NaClO4 were investigated. The systems were characterized by the measurements of differential capacity, zero charge potential and surface tension at this potential. The data were analyzed to obtain the surface pressure and relative surface excess of thiourea, m-toluidine or p-toluidine as a function of charge and bulk concentration of these substances. The standard Gibbs energy of adsorption and parameters α, B obtained from the Frumkin and virial isotherms were compared. The electrostatic parameters of the inner layer were determined.

Singular sets of anisotropic weighted capacity zero and degenerate quasilinear parabolic equations

The aim of this paper is to characterise sets of anisotropic weighted capacity zero. In this we generalise previous known results for the isotropic equivalent. A particular case of this zero capacity set is used to generalise removable singularity results for weak solutions of degenerate quasilinear parabolic equations and for their elliptic equivalent when its structure is still essentially isotropic, with the anisotropy confined to the mixed norms of the generalised Lebesgue spaces involved.