This work investigates nonlocal boundary-value problems in which the primary operator and the operators within the boundary conditions encompass differential operators and twisting transformations. These problems extend classical boundary-value problem frameworks by incorporating nonlocal effects and complex interactions introduced through twisting operators. A novel definition of trajectory symbols is presented, tailored to address the unique characteristics of this class of problems. We demonstrate that the elliptic nature of these problems ensures the generation of Fredholm operators in the appropriate Sobolev spaces. This result establishes well-posedness, confirming that the problems admit solutions that depend continuously on the given data. Furthermore, we provide a comprehensive ellipticity condition specific to these nonlocal problems, serving as a key criterion for their solvability. The findings expand the theory of elliptic boundary-value problems by including settings with nonlocal and shear-like operators, offering insights into their analytical structure. This work contributes to the broader understanding of boundary-value problems with complex operator interactions and provides a foundation for future investigations into more generalized operator frameworks and applications.
In this article, we extend and refine the work of Mitidieri and Pohozhaev concerning the nonexistence of nontrivial weak solutions to nonlinear inequalities and systems. Specifically, we focus on problems involving integer powers of the Laplace operator and nonlinear terms of the form a(x)∣∇(Δmu)∣q+b(x)∣∇u∣sa(x) |\nabla(\Delta^m u)|^q + b(x) |\nabla u|^s. By leveraging the nonlinear capacity method and carefully selecting suitable test functions, we derive optimal a priori estimates. These estimates allow us to prove, via contradiction, the nonexistence of nontrivial weak solutions to the given nonlinear inequalities and systems. Our results provide new insights into the conditions under which solutions fail to exist, contributing to the broader understanding of nonlinear partial differential equations with gradient nonlinearity.
In this study, we explore the geometric properties of spheres on a compact hypersurface in Euclidean space Rn+1\mathbb{R}^{n+1} using the support function θ\theta and the tangential component ψT\psi_T of the position vector field ψ\psi. The first characterization extends existing results by removing constraints on the tangential component ψT\psi_T, and applies alternative proof techniques to obtain new insights. In the second characterization, we focus on the specific role of the support function θ\theta in defining these geometric properties, providing a more comprehensive understanding of the structure of spheres within the hypersurface. This work contributes to the broader field of differential geometry by offering novel approaches to the study of hypersurfaces and their associated curvature characteristics.
In this paper, we investigate prime filters and minimal prime filters on a paradistributive latticoid (PDL) and establish significant theoretical results. We demonstrate that the annihilator filter S∙S^\bullet is equal to the intersection of all prime filters not containing SS, providing a novel perspective on filter structures in PDLs. Furthermore, we explore minimal prime filters, analyzing their unique properties and relationships within the PDL framework. Our study also presents equivalent conditions for a PDL to be relatively complemented, offering insights into its algebraic structure. Additionally, we derive and discuss the topological properties of the spaces of prime filters and minimal prime filters, emphasizing their interactions and contributions to the broader understanding of PDLs. These results expand the foundational knowledge of PDLs and provide new tools for analyzing their algebraic and topological characteristics.
This paper develops a mathematical model of plasma transfer in an open magnetic trap, incorporating the boundary condition of zero plasma concentration at infinity. Experimental data from the SMOLA trap at the Budker Institute of Nuclear Physics SB RAS serve as validation for the model. Plasma confinement is achieved by transferring momentum from a helical-symmetric magnetic field to a rotating plasma. The model is formulated using a stationary plasma transfer equation in an axially symmetric configuration, incorporating second-order spatial derivatives. An optimal numerical template for approximating mixed derivatives is determined through a benchmark test problem. The numerical implementation of the model is evaluated by comparing the establishment method and the successive over-relaxation method. The findings demonstrate the model’s capability to accurately describe plasma dynamics in helical magnetic fields and its effectiveness in optimizing plasma confinement.