THEORY OF SURFACES IN FOUR-DIMENSIONAL GALILEAN SPACE

By means of a special choice of coordinate lines of the surface in four-dimensional Galilean space, the first and second quadratic shape of the surface is defined. It has been proved that the second-order surface equation in three-dimensional space can be converted to a canonical form by means of a special transformation, which is the rotation of the coordinate axes of three-dimensional Galilean space. Furthermore, the transformation matrix is an element of the Heisenberg group that is neither symmetric nor orthogonal. In four-dimensional space R41 - the concept of a surface indicator is introduced and the main curvature of the surface is defined.

Symmetric solutions of the singular minimal surface equation

We classify all rotational symmetric solutions of the singular minimal surface equation in both cases $$\alpha <0$$ α < 0 and $$\alpha >0$$ α > 0 . In addition, we discuss further geometric and analytic properties of the solutions, in particular stability, minimizing properties and Bernstein-type results.

A Spin-Perturbation for Minimal Surfaces

We investigate the coupling of the minimal surface equation with a spinor of harmonic type. This arises as the Euler–Lagrange equations of the sum of the volume functional and the Dirac action, defined on an appropriated Dirac bundle. The solutions show a relation to Dirac-harmonic maps under some constraints on the energy-momentum tensor, extending the relation between Riemannian minimal surface and harmonic maps.

On Convergence of Polynomial Approximate Solutions of Minimal Surface Equations in Domains Satisfying the Cone Condition

In this paper we consider the polynomial approximate solutions of the minimal surface equation. It is shown that under certain conditions on the geometric structure of the domain the absolute values of the gradients of the solutions are bounded as the degree of these polynomials increases. The obtained properties imply the uniform convergence of approximate solutions to the exact solution of the minimal surface equation. In numerical solving of boundary value problems for equations and systems of partial differential equations, a very important issue is the convergence of approximate solutions.The study of this issue is especially important for nonlinear equations since in this case there is a series of difficulties related with the impossibility of employing traditional methods and approaches used for linear equations. At present, a quite topical problem is to determine the conditions ensuring the uniform convergence of approximate solutions obtained by various methods for nonlinear equations and system of equations of variational kind (see, for instance, [1]). For nonlinear equations it is first necessary to establish some a priori estimates of the derivatives of approximate solutions.In this paper, we gave a substantiation of the variational method of solving the minimal surface equation in the case of multidimensional space.We use the same approach that we used in [3] for a two-dimensional equation. Note that such a convergence was established in [3] under the condition that a certain geometric characteristic Δ(Ω) in the domain Ω, in which the solutions are considered, is positive. In particular, domain with a smooth boundary satisfied this requirement. However, this characteristic is equal to zero for a fairly wide class of domains with piecewise-smooth boundaries and sufficiently "narrow" sections at the boundary. For example, such a section of the boundary is the vertex of a cone with an angle less than π/2. In this paper, we present another approach to determining the value of Δ(Ω) in terms of which it is possible to extend the results of the work [3] in domain satisfying cone condition.

CALCULATION OF PROFILED FLOORING USING THE MID-SURFACE EQUATION OF THE ORTHOTROPIC PLATE

Стальной профилированный настил является конструктивно-ортотропным материалом, поскольку его ортотропия обусловлена не свойствами материала, а формой поперечного сечения получаемой на этапе производства. Учет ортотропных свойств профилированного настила во время расчета производится крайне редко, поскольку является трудоемким процессом. Предложена методика расчета профилированного настила с учетом его ортотропных свойств, реализуемая в различных расчетных программных комплексах. Приведены результаты расчетов в пакете Matlab и ПК Лира-САПР.