A Variational Formulation for Dirac Operators in Bounded Domains. Applications to Spectral Geometric Inequalities

SynopsisAfter a study made for bounded domains and in the periodic case, we investigate the variational formulation of Schrödinger-Poisson systems set on the whole space ℝd,d≦ 3. This variational formulation leads to a uniqueness result, while the existence of a solution is proved only for ‘small data’ because of the lack of coerciveness. The end of this paper briefly presents the extension of this formalism to a physically relevant problem where the potential is periodic in one direction.

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Multiple Solitary Waves For a Non-homogeneous Schrödinger–Maxwell System in ℝ3

Advanced Nonlinear Studies ◽

10.1515/ans-2006-0203 ◽

2006 ◽

Vol 6 (2) ◽

Cited By ~ 23

Author(s):

A. Salvatore

Keyword(s):

Eigenvalue Problem ◽

Solitary Waves ◽

Variational Formulation ◽

Maxwell’S Equations ◽

Schrodinger Equation ◽

Maxwell's Equations ◽

Standing Waves ◽

Bounded Domains ◽

Maxwell System ◽

Fibering Method

AbstractWe look for standing waves of nonlinear Schrödinger equationcoupled with Maxwell’s equations. We use the variational formulation introduced by Benci and Fortunato in 1992 for studying an eigenvalue problem for the Schrödinger-Maxwell system in bounded domains. We establish the existence of multiple standing waves both in the homogeneous and the non-homogeneous cases by means of the fibering method introduced by Pohozaev.

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Combining the DPG Method with Finite Elements

Computational Methods in Applied Mathematics ◽

10.1515/cmam-2017-0041 ◽

2018 ◽

Vol 18 (4) ◽

pp. 639-652

Author(s):

Thomas Führer ◽

Norbert Heuer ◽

Michael Karkulik ◽

Rodolfo Rodríguez

Keyword(s):

Variational Formulation ◽

Model Problem ◽

Standard Test ◽

Well Posedness ◽

Bounded Domains ◽

Test Functions ◽

Test Space ◽

Optimal Test ◽

Weak Part ◽

Convergence Orders

AbstractWe propose and analyze a discretization scheme that combines the discontinuous Petrov–Galerkin and finite element methods. The underlying model problem is of general diffusion-advection-reaction type on bounded domains, with decomposition into two sub-domains. We propose a heterogeneous variational formulation that is of the ultra-weak (Petrov–Galerkin) form with broken test space in one part, and of Bubnov–Galerkin form in the other. A standard discretization with conforming approximation spaces and appropriate test spaces (optimal test functions for the ultra-weak part and standard test functions for the Bubnov–Galerkin part) gives rise to a coupled DPG-FEM scheme. We prove its well-posedness and quasi-optimal convergence. Numerical results confirm expected convergence orders.

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