The Electromagnetic Field outside the Steadily Rotating Relativistic Uniform System

Using the method of retarded potentials, approximate formulae are obtained that describe the electromagnetic field outside the relativistic uniform system in the form of a charged sphere rotating at a constant speed. For the near, middle and far zones, the corresponding expressions are found for the scalar and vector potentials, as well as for the electric and magnetic fields. Then, these expressions are assessed for correspondence to the Laplace equations for potentials and fields. One of the purposes is to test the truth of the assumption that the scalar potential and the electric field depend neither on the value of the angular velocity of rotation of the sphere nor on the direction to the point where the field is measured. However, calculations show that potentials and fields increase as the observation point gets closer to the sphere’s equator and to the sphere’s surface, compared with the case for a stationary sphere. In this case, additions are proportional to the square of the angular velocity of rotation and the square of the sphere’s radius and inversely proportional to the square of the speed of light. The largest found relative increase in potentials and fields could reach the value of 4% for the rapidly rotating neutron star PSR J1614-2230, if the star were charged. For a proton, a similar increase in fields on its surface near the equator reaches 54%. Keywords: Electromagnetic field, Relativistic uniform system, Rotation.

Tides and Tidal Currents—Guidelines for Site and Energy Resource Assessment

The main aim of this paper was to classify and to analyze the expeditious resource assessment procedure to help energy planners and system designers dealing with tides and tidal currents. Depending on the geographical features of the site to be evaluated, this paper reported the easiest methods to adopt for later working plans, crucial for preliminary considerations but to be supported by in situ measurements and by a more complex and detailed modelling. While tide trends are predictable by using Laplace equations and Fourier series, tidal currents velocities prediction is not easy, requiring suitable methods or hydraulic applications. Natural and artificial sites were analyzed and the best method for each type of them was presented. The latter together highlighting the minimum set of required information was discussed and provided as a toolkit for assessing tides and tidal current energy potential.

A WEIGHTED APPROACH FOR CACCIOPOLI INEQUALITY FOR SOLUTIONS TO P-LAPLACE EQUATIONS

Không gian Sobolev cấp phân số có trọng có nhiều ứng dụng trong phương trình đạo hàm riêng. Trong bài báo này, chúng tôi khảo sát lớp không gian Sobolev cấp phân số có trọng, ứng với hàm trọng là hàm khoảng cách đến biên của miền xác định. Lớp không gian này được sử dụng để thu được một dạng bất đẳng thức dạng Cacciopoli có trọng cho bài toán p-Laplace với dữ liệu độ đo. Kết quả của chúng tôi là mở rộng của bất đẳng thức Cacciopoli trong bài báo gần đây (Tran & Nguyen, 2021b).

On n-superharmonic functions and some geometric applications

In this paper we study asymptotic behaviors of n-superharmonic functions at singularity using the Wolff potential and capacity estimates in nonlinear potential theory. Our results are inspired by and extend [6] of Arsove–Huber and [63] of Taliaferro in 2 dimensions. To study n-superharmonic functions we use a new notion of thinness in terms of n-capacity motivated by a type of Wiener criterion in [6]. To extend [63], we employ the Adams–Moser–Trudinger’s type inequality for the Wolff potential, which is inspired by the inequality used in [15] of Brezis–Merle. For geometric applications, we study the asymptotic end behaviors of complete conformally flat manifolds as well as complete properly embedded hypersurfaces in hyperbolic space. These geometric applications seem to elevate the importance of n-Laplace equations and make a closer tie to the classic analysis developed in conformal geometry in general dimensions.