Here, ∆ is the Laplacian, μ ∈ R is a fixed parameter, ν is the outward unit normal to Ω, and Ω+ := Ω, Ω− := Rn \ Ω̄. For 1 < p < ∞, L̇p1(∂Ω) is the classical homogeneous Lp-based Sobolev spaces of order one on ∂Ω, M denotes the non-tangential maximal operator, ∂ν is the normal derivative and all restrictions to the boundary are taken in the non-tangential limit sense; detailed definitions are giv...

We study the following p-Laplacian equation with nonlinear boundary conditions: -Δ(p)u + μ(x)|u|(p-2)u = f(x,u) + g(x,u),x ∈ Ω, | ∇u|(p-2)∂u/∂n = η|u|(p-2)u and x ∈ ∂Ω, where Ω is a bounded domain in ℝ(N) with smooth boundary ∂Ω. We prove that the equation has infinitely many weak solutions by using the variant fountain theorem due to Zou (2001) and f, g do not need to satisfy the (P.S) or (P.S...

Consider a nonlinear wave equation for massless scalar field with self-interaction in the spatially flat de Sitter spacetime. We show that blow-up finite time occurs arbitrary power nonlinearity as well upper bounds of lifespan solutions. The condition is same accelerated expanding Friedmann–Lemaître–Robertson–Walker (FLRW) also results space derivative term.

We revisit the Hartree-Fock (HF) calculation for the uniform electron gas, or jellium model, whose predictions--divergent derivative of the energy dispersion relation and vanishing density of states (DOS) at the Fermi level--are in qualitative disagreement with experimental evidence for simple metals. Currently, this qualitative failure is attributed to the lack of screening in the HF equations...

In this study, a derivative method is applied for the determination of dye concentration in fabrics dyed with tricomponent dye mixtures. The reflectance of spectra of the samples were recorded between 400 and 700 nm. Initially, the obtained (K/S) spectra were divided by the spectrum of a standard sample dyed with two of the components in the ternary mixture and the derivative spectra were calcu...

The behavior of the average velocity for a classical particle in the one-dimensional Fermi accelerator model under sawtooth external force is considered. For elastic collisions, it is known that the average velocity of the particle grows unlimitedly because of the discontinuities of the derivative of the moving wall's position with respect to time. However, and contrary to what was expected to ...

We present here results on the fine structure of the static qq̄ potential in d = 4 SU(3) Yang-Mills theory. The potential is obtained from Polyakov loop correlators having separations between 0.3 and 1.2 fermi. Measurements were carried out on lattices of spatial extents of about 4 and 5.4 fermi. The temporal extent was 5.4 fermi in both cases. The results are analyzed in terms of the force betw...

We discuss a recently proposed analytic solution to the Thomas– Fermi (TF) equation and show that earlier approaches provide more accurate results. In particular, we show that a simple and straightforward rational approximation to the TF equation yields the slope at origin with unprecedented accuracy, as well as remarkable values of the TF function and its first derivative for other coordinate ...