General 3D Analytical Method for Eddy-Current Coupling with Halbach Magnet Arrays Based on Magnetic Scalar Potential and H-Functions

Rapid and accurate eddy-current calculation is necessary to analyze eddy-current couplings (ECCs). This paper presents a general 3D analytical method for calculating the magnetic field distributions, eddy currents, and torques of ECCs with different Halbach magnet arrays. By using Fourier decomposition, the magnetization components of Halbach magnet arrays are determined. Then, with a group of H-formulations in the conductor region and Laplacian equations with magnetic scalar potential in the others, analytical magnetic field distributions are predicted and verified by 3D finite element models. Based on Ohm’s law for moving conductors, eddy-current distributions and torques are obtained at different speeds. Finally, the Halbach magnet arrays with different segments are optimized to enhance the fundamental amplitude and reduce the harmonic contents of air-gap flux densities. The proposed method shows its correctness and validation in analyzing and optimizing ECCs with Halbach magnet arrays.

Multiplicity of solutions for Robin problem involving the p(x)-laplacian

This paper is concerned with the existence and multiplicity of solutions for p(x)-Laplacian equations with Robin boundary condition. Our technical approach is based on variational methods.

Global existence and asymptotic behavior of solutions to fractional ( p , q )-Laplacian equations

The aim of this paper is to consider the following fractional parabolic problem u t + ( − Δ ) p α u + ( − Δ ) q β u = f ( x , u ) ( x , t ) ∈ Ω × ( 0 , ∞ ) , u = 0 ( x , t ) ∈ ( R N ∖ Ω ) × ( 0 , ∞ ) , u ( x , 0 ) = u 0 ( x ) x ∈ Ω , where Ω ⊂ R N is a bounded domain with Lipschitz boundary, ( − Δ ) p α is the fractional p-Laplacian with 0 < α < 1 < p < ∞, ( − Δ ) q β is the fractional q-Laplacian with 0 < β < α < 1 < q < p < ∞, r > 1 and λ > 0. The global existence of nonnegative solutions is obtained by combining the Galerkin approximations with the potential well theory. Then, by virtue of a differential inequality technique, we give a decay estimate of solutions.

A simple proof of the generalized Leibniz rule on bounded Euclidean domains

This paper is devoted to a simple proof of the generalized Leibniz rule in bounded domains. The operators under consideration are the so-called spectral Laplacian and the restricted Laplacian. Equations involving such operators have lately been considered by Constantin and Ignatova in the framework of the SQG equation [P. Constantin and M. Ignatova, Critical SQG in bounded domains, Ann. PDE 2 2016, 2, Article ID 8] in bounded domains, and by two of the authors [Q.-H. Nguyen and J. L. Vázquez, Porous medium equation with nonlocal pressure in a bounded domain, Comm. Partial Differential Equations 43 2018, 10, 1502–1539] in the framework of the porous medium with nonlocal pressure in bounded domains. We will use the estimates in this work in a forthcoming paper on the study of porous medium equations with pressure given by Riesz-type potentials.

Four Solutions for Fractional p-Laplacian Equations with Asymmetric Reactions

We consider a Dirichlet type problem for a nonlinear, nonlocal equation driven by the degenerate fractional p-Laplacian, whose reaction combines a sublinear term depending on a positive parameter and an asymmetric perturbation (superlinear at positive infinity, at most linear at negative infinity). By means of critical point theory and Morse theory, we prove that, for small enough values of the parameter, such problem admits at least four nontrivial solutions: two positive, one negative, and one nodal. As a tool, we prove a Brezis-Oswald type comparison result.

Compact Sobolev-Slobodeckij embeddings and positive solutions to fractional Laplacian equations

In this work, we study the existence of a positive solution to an elliptic equation involving the fractional Laplacian (−Δ) s in ℝ n , for n ≥ 2, such as (0.1) ( − Δ ) s u + E ( x ) u + V ( x ) u q − 1 = K ( x ) f ( u ) + u 2 s ⋆ − 1 . $$(-\Delta)^{s} u+E(x) u+V(x) u^{q-1}=K(x) f(u)+u^{2_{s}^{\star}-1}.$$ Here, s ∈ (0, 1), q ∈ 2 , 2 s ⋆ $q \in\left[2,2_{s}^{\star}\right)$ with 2 s ⋆ := 2 n n − 2 s $2_{s}^{\star}:=\frac{2 n}{n-2 s}$ being the fractional critical Sobolev exponent, E(x), K(x), V(x) > 0 : ℝ n → ℝ are measurable functions which satisfy joint “vanishing at infinity” conditions in a measure-theoretic sense, and f (u) is a continuous function on ℝ of quasi-critical, super-q-linear growth with f (u) ≥ 0 if u ≥ 0. Besides, we study the existence of multiple positive solutions to an elliptic equation in ℝ n such as (0.2) ( − Δ ) s u + E ( x ) u + V ( x ) u q − 1 = λ K ( x ) u r − 1 , $$(-\Delta)^{s} u+E(x) u+V(x) u^{q-1}=\lambda K(x) u^{r-1},$$ where 2 < r < q < ∞(both possibly (super-)critical), E(x), K(x), V(x) > 0 : ℝ n → ℝ are measurable functions satisfying joint integrability conditions, and λ > 0 is a parameter. To study (0.1)-(0.2), we first describe a family of general fractional Sobolev-Slobodeckij spaces Ms ;q,p (ℝ n ) as well as their associated compact embedding results.