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The concentration-compactness principle in the calculus of variations. The limit case, Part II.

Pierre-Louis Lions (1985)

Revista Matemática Iberoamericana

This paper is the second part of a work devoted to the study of variational problems (with constraints) in functional spaces defined on domains presenting some (local) form of invariance by a non-compact group of transformations like the dilations in RN. This contains for example the class of problems associated with the determination of extremal functions in inequalities like Sobolev inequalities, convolution or trace inequalities... We show how the concentration-compactness principle and method...

The concentration-compactness principle in the calculus of variations. The limit case, Part I.

Pierre-Louis Lions (1985)

Revista Matemática Iberoamericana

After the study made in the locally compact case for variational problems with some translation invariance, we investigate here the variational problems (with constraints) for example in RN where the invariance of RN by the group of dilatations creates some possible loss of compactness. This is for example the case for all the problems associated with the determination of extremal functions in functional inequalities (like for example the Sobolev inequalities). We show here how the concentration-compactness...

The Curvature of a Set with Finite Area

Elisabetta Barozzi (1994)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In a paper, by myself, E. Gonzalez and I. Tamanini (see [2]), it was proven that all sets of finite perimeter do have a non trivial variational property, connected with the mean curvature of their boundaries. In the present article, that variational property is made more precise.

The Dirichlet problem with sublinear nonlinearities

Aleksandra Orpel (2002)

Annales Polonici Mathematici

We investigate the existence of solutions of the Dirichlet problem for the differential inclusion 0 ∈ Δ x ( y ) + ∂ x G ( y , x ( y ) ) for a.e. y ∈ Ω, which is a generalized Euler-Lagrange equation for the functional J ( x ) = ∫ Ω 1 / 2 | ∇ x ( y ) | ² - G ( y , x ( y ) ) d y . We develop a duality theory and formulate the variational principle for this problem. As a consequence of duality, we derive the variational principle for minimizing sequences of J. We consider the case when G is subquadratic at infinity.

The distance between subdifferentials in the terms of functions

Libor Veselý (1993)

Commentationes Mathematicae Universitatis Carolinae

For convex continuous functions f , g defined respectively in neighborhoods of points x , y in a normed linear space, a formula for the distance between ∂ f ( x ) and ∂ g ( y ) in terms of f , g (i.eẇithout using the dual) is proved. Some corollaries, like a new characterization of the subdifferential of a continuous convex function at a point, are given. This, together with a theorem from [4], implies a sufficient condition for a family of continuous convex functions on a barrelled normed linear space to be locally uniformly...

The double bubble conjecture.

Hass, Joel, Hutchings, Michael, Schlafly, Roger (1995)

Electronic Research Announcements of the American Mathematical Society [electronic only]

Currently displaying 21 – 40 of 272