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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 current situation in the linear problem of Molodenskii

Fausto Sacerdote, Fernando Sansò (1984)

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

Si prova l'esistenza di un'unica soluzione debole che dipende con continuità dai dati al contorno per il problema lineare di Molodenskii in approssimazione quasi sferica, nel caso che la superficie al contorno soddisfi una condizione di cono. Si segue un approccio costruttivo diretto, che generalizza una procedura precedentemente elaborata per il problema semplice di Molodenskii. Inoltre si prova che la soluzione ha derivate prime a quadrato integrabile al contorno, il che è essenziale per le applicazioni...

The density of solenoidal functions and the convergence of a dual finite element method

Ivan Hlaváček (1980)

Aplikace matematiky

A proof is given of the following theorem: infinitely differentiable solenoidal vector - functions are dense in the space of functions, which are solenoidal in the distribution sense only. The theorem is utilized in proving the convergence of a dual finite element procedure for Dirichlet, Neumann and a mixed boundary value problem of a second order elliptic equation.

The fascinating homotopy structure of Sobolev spaces

Haïm Brezis (2003)

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

We discuss recent developments in the study of the homotopy classes for the Sobolev spaces W 1 , p ⁢ M ; N . In particular, we report on the work of H. Brezis - Y. Li [5] and F.B. Hang - F.H. Lin [9].

The form boundedness criterion for the relativistic Schrödinger operator

Vladimir Maz'ya, Igor Verbitsky (2004)

Annales de l’institut Fourier

We establish necessary and sufficient conditions on the real- or complex-valued potential Q defined on ℝ n for the relativistic Schrödinger operator - Δ + Q to be bounded as an operator from the Sobolev space W 2 1 / 2 ( ℝ n ) to its dual W 2 - 1 / 2 ( ℝ n ) .

The fractional integral between weighted Orlicz and B M O φ spaces on spaces of homogeneous type

Gladis Pradolini, Oscar Salinas (2003)

Commentationes Mathematicae Universitatis Carolinae

In this work we give sufficient and necessary conditions for the boundedness of the fractional integral operator acting between weighted Orlicz spaces and suitable B M O φ spaces, in the general setting of spaces of homogeneous type. This result generalizes those contained in [P1] and [P2] about the boundedness of the same operator acting between weighted L p and Lipschitz integral spaces on ℝ n . We also give some properties of the classes of pairs of weights appearing in connection with this boundedness.

The general form of local bilinear functions

Milan Práger (1993)

Applications of Mathematics

The scalar product of the FEM basis functions with non-intersecting supports vanishes. This property is generalized and the concept of local bilinear functional in a Hilbert space is introduced. The general form of such functionals in the spaces L 2 ( a , b ) and H 1 ( a , b ) is given.

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