The article focuses on differential geometry of ρ-dimensional complexes C^ρ of m-dimensional planes in the projective space Pⁿ that contain a finite number of developable surfaces for which n − m different developable surfaces have one common (m+1)-dimensional tangent plane to the developable surface. At the same time, the same n−m different developable surfaces have one common characteristic (m−1)-dimensional plane common for two infinitely close generatrices of the developable surface. This article relates to researches on projective differential geometry based on the Cartan moving frame method and the method of exterior differential forms. These methods make it possible to study the differential geometry of submanifolds of different dimensions of a Grassmann manifold from a single viewpoint, as well as to extend the results to wider classes of manifolds of multidimensional planes. To study such submanifolds, we apply the Grassmann map of the manifold G(m,n) onto the (m + 1)(n − m)-dimensional algebraic manifold (m,n) of the space P^N, where N 

We study the Dirichlet problem for the composite type differential equations D_t[(−1)^pD_t^2p+1u−h(x)u_xx] +a(x)u_xx +c(x,t)u = f(x,t) in the domain Q = {(x,t) : x ∈ (−1,0) ∪ (0,1), t ∈ (0,T), 0 < T < +∞}, where p ≥1 is an integer, D_t^k = ∂^k/∂t^k , and D_t = ∂/∂t . The feature of such quations is that the coefficients h(x) and a(x) can have a discontinuity of the first kind when passing through the point x = 0. In addition to the usual Dirichlet boundary conditions, the problem under study also specifies the conjugation conditions on the line x = 0. Existence and uniqueness theorems are proved for regular solutions (those having all generalized Sobolev derivatives).

A metric connection with vectorial torsion, or a semi-symmetric metric connection, was discovered by E. Cartan. Later, many mathematicians studied the properties of this connection. For example, K. Yano, I. Agricola and other mathematicians investigated the properties of the curvature tensor, geodesic lines, and also the behavior of the connection under conformal deformations of the original metric.

In this paper, we study the Einstein equation on three-dimensional locally homogeneous (pseudo)Riemannian manifolds with metric connection with invariant vectorial torsion. A theorem is obtained stating that all such manifolds are either Einstein manifolds with respect to the Levi-Civita connection or conformally flat. Earlier, the Einstein equation in the case of three-dimensional locally symmetric (pseudo)Riemannian manifolds have been investigated by the authors. 

We study the solvability of boundary value problems for some classes of degenerate quasi-elliptic equations. The main feature of the problems under study is that, despite the degeneration, boundary conditions should still be imposed on the boundary manifolds. We prove the existence and uniqueness theorems for the regular solutions, those having all generalized Sobolev derivatives required in the equation in the inner subdomains. Moreover, we describe some possible enhancements and generalizations of the obtained results.

Mathematical modeling and research of problems on the deformation of inhomogeneous bodies containing cracks along elastic inclusions involves setting the conjugation conditions at the interface between different materials. Difficulties are associated with the possibility of large stress values appearing near the inclusions. Determining the inhomogeneous bodies with the most optimal parameters is one of the most popular areas of theoretical and experimental research. In this paper, we study the problem of optimal control of the angle of the crack inclination to the median plane in the equilibrium problem for an elastic Timoshenko plate containing an oblique crack at the boundary of an elastic inclusion. The complexity of the problem is due to the fact that the non-penetration condition on the crack faces is given in the form of inequality. The quality functional characterizes the deviation from the specified displacements. We prove solvability of the optimal control problem and establish continuous dependence of the solutions on the value of the crack inclination angle. 

The paper provides some of the results that were presented at the IX International Conference on Mathematical Modeling dedicated to the 75th anniversary of V. N. Vragov and are related to the study of problems with an unknown boundary by methods of compactness. A substantiation of the theorem on relative compactness is given, which can be used in the study of problems of the Stefan type with an unknown part of the boundary, as well as in problems for equations of variable type with an unknown boundary of type change. An example of such problem is given, and it is shown that the estimates obtained for approximate solutions of the equation and functions describing a sequence of boundaries approaching the sought boundary of the phase transition completely coincide with the conditions formulated in the main theorem on compactness.

The author is not aware of theorems of such type. The theorem is a new kind of compactness theorem, adapted to problems of the Stefan type.

For better perception, the simplest conditions are given under which the result is valid, which coincide with the conditions of the considered example. However, the result can be generalized to much more general situations, including the number of phase transition boundaries and replacing the estimate of the second derivative with an estimate of a more complex aggregate that occurs in equations with degenerations on solutions.

We study the solvability in anisotropic Sobolev spaces of nonlocal in time problems for the differential equations of composite (Sobolev) type u_tt + (α∂/∂t +β) Δu+γu=f(x,t), where x = (x₁,...,x_n) ∈ Ω  ⊂Rⁿ, t ∈ (0,T), 0 < T < +∞, α, β, and γ are real numbers, and f(x,t) is a given function. We prove theorems of existence and non-existence, uniqueness and non-uniqueness for regular solutions, those having all generalized Sobolev derivatives in the equation.