The concentration-compactness principle in the calculus of variations. The locally compact case. I
The concentration-compactness principle in the calculus of variations. The locally compact case. I
This paper presents a general method - called concentration-compactness method - for solving certain minimization problems on unbounded domains. This method applies to problems with some form of local compactness. For minimization problems with constraints, sub-additivity inequalities are obtained for the infimum of the problem considered as a function of the value of the constraint. The concentration-compactness method states that ''all minimizing sequences are relatively compact if and only if the sub- additivity inequalities are strict.'' This principle is applied to various examples - rotating stars problem, Choquard-Pekar problem, and nonlinear fields equations.
Choquard- Pekar problem, Local compactness, \(\sigma\)-compactness, Methods involving semicontinuity and convergence; relaxation, Equations involving nonlinear operators (general), Hartree-Fock problems, Schrödinger equations, Existence theories for optimal control problems involving partial differential equations, rotating stars, Variational principles in infinite-dimensional spaces, minimization problems on unbounded domains, Choquard-Pekar problem, nonlinear fields equations, minimization over manifolds, Nonlinear boundary value problems for linear elliptic equations, local compactness, Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics, concentration-compactness, Existence theories for problems in abstract spaces, concentration-compactness principle, rotating star problem, Existence theories for free problems in two or more independent variables, Variational principles of physics, Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems, nonlinear field equations
Choquard- Pekar problem, Local compactness, \(\sigma\)-compactness, Methods involving semicontinuity and convergence; relaxation, Equations involving nonlinear operators (general), Hartree-Fock problems, Schrödinger equations, Existence theories for optimal control problems involving partial differential equations, rotating stars, Variational principles in infinite-dimensional spaces, minimization problems on unbounded domains, Choquard-Pekar problem, nonlinear fields equations, minimization over manifolds, Nonlinear boundary value problems for linear elliptic equations, local compactness, Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics, concentration-compactness, Existence theories for problems in abstract spaces, concentration-compactness principle, rotating star problem, Existence theories for free problems in two or more independent variables, Variational principles of physics, Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems, nonlinear field equations
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