We consider the first mixed problem for the system of Vlasov-Poisson equations with a given external magnetic field in a bounded domain. This problem describes the kinetics of high-temperature plasma in controlled thermonuclear fusion plants and is considered with respect to unknown functions: electric field potential, distribution functions of positively charged ions and electrons. Additionally, we assumed that the distribution functions of charged particles satisfy the condition of mirror reflection from the boundary of the domain under consideration. We prove the existence of global weak solutions of such a problem.

We consider the Vlasov–Poisson system with external magnetic field in a half-space with the Neumann boundary condition for the electric potential and specular reflection on a boundary. For arbitrary compactly supported initial density distribution functions, we obtain sufficient conditions for external magnetic field, which provide global existence of density distribution functions with compact supports lying at some distance from a boundary.

We consider the damping problem for a nonstationary control system described by a system of differential-difference equations of neutral type with smooth matrix coefficients and several delays. A connection has been established between the variational problem corresponding to the problem of calming a system with delay and the boundary value problem for a system of second-order differential equations. A priori estimates of solutions are obtained. A theorem on the solvability of the considered boundary value problem is proved.

We consider an ordinary fourth-order differential equation with a spectral parameter and integral conditions containing a linear combination of derivatives of an unknown function. In terms of equivalent norms, a priori estimates for solutions of this problem are obtained for sufficiently large values of the spectral parameter. Using these estimates, the discreteness, the sectorial structure of spectrum, and the Fredholm solvability of problem are proven.

We consider the first mixed problem for the Vlasov–Poisson system with an external magnetic field in a domain with piecewise smooth boundary. This problem describes the kinetics of a two-component high-temperature plasma under the influence of a self-consistent electric field and an external magnetic field. The existence of global weak solutions is proved. In the case of a cylindrical domain, sufficient conditions are obtained for the existence of global weak solutions with supports in a strictly internal cylinder; this corresponds to the confinement of high-temperature plasma in a mirror trap.

Рассматривается задача на собственные функции и собственные значения для дифференциально-разностных операторов. Получены необходимые и достаточные условия сохранения гладкости обобщенных собственных функций на всем интервале. Приводится пример дифференциально-разностного оператора, имеющего счетное множество собственных функций, гладкость которых нарушается внутри интервала, и счетное множество собственных функций, гладкость которых сохраняется. Библиография: 14 названий.

The eigenfunction–eigenvalue problem for differential–difference operators is considered. Necessary and sufficient conditions for preserving the smoothness of generalized eigenfunctions over the entire interval are obtained. An example is given of a differential–difference operator having a countable set of eigenfunctions whose smoothness is violated inside the interval and a countable set of eigenfunctions whose smoothness is preserved.

We consider the first mixed problem for the Vlasov–Poisson system with a homogeneous external magnetic field in an infinite cylinder. For solutions with supports of the distribution density functions of charged particles lying strictly in the inner cylinder, an a priori estimate of the strength of the self-consistent electric field via the initial distribution density functions is obtained.

Рассматривается вторая краевая задача для дифференциально-разностного уравнения второго порядка с переменными коэффициентами на интервале $(0,d)$. Исследован вопрос о том, какие условия на правую часть уравнения обеспечивают гладкость обобщенных решений краевой задачи на всем интервале $(0,d)$ при $d \notin \mathbb{N}$. Библиография: 13 названий.

Three-Dimensional Stationary Spherically Symmetric Stellar Dynamic Models Depending on the Local Energy. Juergen Batt, Enno Joern, Alexander L. Skubachevskii The stellar dynamic models considered here are triples (f,rho,U) of three functions: the distribution function f=f(r,u), the local density rho=rho(r) and the Newtonian potential U=U(r), where r:=|x|, u:=|v| ((x,v) in R^3xR^3 are the space-velocity coordinates), and f is a function q of the local energy E=U(r)+u^2/2. Our first result is an answer to the following question: Given a (positive) function p=p(r) on a bounded interval [0,R], how can one recognize p as the local density of a stellar dynamic model of the given type ("inverse problem")? If this is the case, we say that p is "extendable" (to a complete stellar dynamic model). Assuming that p is strictly decreasing, we reveal the connection between p and F, which appears in the nonlinear integral equation p=FU[p] and the solvability of Eddington's equation between F and q. Second, we investigate the following question ("direct problem"): Which q induce distribution functions f of the form f=q(-E(r,u)-E0) of a stellar dynamic model? This leads to the investigation of the nonlinear equation p=FU[p] in an approximate and constructive way by mainly numerical methods. The paper extends preceding work on flat galaxies to the three-dimensional case. In particular, the present answer to the extendability problem is completely different as in [1]. The present paper also opens the way to further explicit solutions of the Vlasov-Poisson system beyond the classical known examples which are for instant given in [4]. Keywords: Vlasov-Poisson system, stationary solutions, numerical approximation, mathematical physic, galaxy astrophysics.

We consider strongly elliptic differential-difference equations with mixed boundary conditions in a cylindrical domain. We establish the connection between such problems and nonlocal mixed problems for strongly elliptic differential equations, and prove the uniqueness of solutions.

We consider a control system described by a system of differential equations of retardedtype with variable matrix coefficients and several delays. The relationship between the variationalproblem for a nonlocal functional describing the multidimensional control system with delays andthe corresponding boundary value problem for the systems of differential–difference equations isestablished. The existence, uniqueness, and smoothness of the generalized solution of theboundary value problem on the entire interval are proved.

This article deals with mixed boundary value problem for an elliptic differential-difference equation in a cylinder. There are obtained results on the smoothness of generalized solutions of such problem in subdomains, and necessary and sufficient conditions for the preservation of smoothness on the boundaries of neighboring subdomains.

We consider a control system described by the system of differential-difference equations of neutral type with variable matrix coefficients and several delays. We establish the relation between the variational problem for the nonlocal functional describing the multidimensional control system with delays and the corresponding boundary-value problem for the system of differential-difference equations. We prove the existence and uniqueness of the generalized solution of this boundary-value problem.

We address the existence of stationary solutions of the Vlasov-Poisson system on a domain Omega subset of R-3 describing a high-temperature plasma which due to the influence of an external magnetic field is spatially confined to a subregion of Omega. In a first part we provide such an existence result for a generalized system of Vlasov-Poisson type and investigate the relation between the strength of the external magnetic field, the sharpness of the confinement and the amount of plasma that is confined measured in terms of the total charges. The key tools here are the method of sub-/supersolutions and the use of first integrals in combination with cutoff functions. In a second part we apply these general results to the usual Vlasov-Poisson equation in three different settings: the infinite and finite cylinder, as well as domains with toroidal symmetry. This way we prove the existence of stationary solutions corresponding to a two-component plasma confined in a Mirror trap, as well as a Tokamak.