Sobolev embeddings into BMO, VMO, andL ∞

LetX be a rearrangement-invariant Banach function space onR ⁿ and letV ¹ X be the Sobolev space of functions whose gradient belongs toX. We give necessary and sufficient conditions onX under whichV ¹ X is continuously embedded into BMO or intoL _∞. In particular, we show thatL _{n, ∞} is the largest rearrangement-invariant spaceX such thatV ¹ X is continuously embedded into BMO and, similarly,L _{n, 1} is the largest rearrangement-invariant spaceX such thatV ¹ X is continuously embedded intoL _∞. We further show thatV ¹ X is a subset of VMO if and only if every function fromX has an absolutely continuous norm inL _{n, ∞} . A compact inclusion ofV ¹ X intoC ⁰ is characterized as well.     

A quantitative Sobolev inequality in BV

A quantitative version of the standard Sobolev inequality, with sharp constant, for functions u in W1,1(Rn) (or BV(Rn)) is established in terms of a distance of u from the manifold of all multiples of characteristic functions of balls. Inequalities involving non-Euclidean norms of the gradient are discussed as well.     

Concentration-compactness principles for Moser–Trudinger inequalities: new results and proofs

We are concerned with the best exponent in Concentration-Compactness principles for the borderline case of the Sobolev inequality. We present a new approach, which both yields a rigorous proof of the relevant principle in the standard case when functions vanishing on the boundary are considered, and enables us to deal with functions with unrestricted boundary values.     

Relaxation effects in Fe17 and Fe19 oxo-bridged iron clusters

The magnetic properties of a new type of iron clusters are studied by means of Mössbauer spectroscopy in an applied magnetic field. The results obtained agree with the ones obtained from magnetic susceptibility measurements. They are well explained in terms of relaxation determined by fluctuations in the hyperfine field due to spin-spin interaction. The triangular arrangement of the iron ions plays an essential role in this mechanism giving rise to spin frustration effects.     

Sharp Morrey-Sobolev inequalities and the distance from extremals

Quantitative versions of sharp estimates for the supremum of Sobolev functions in , , with remainder terms depending on the distance from the families of extremals, are established.

     

Global Lipschitz Regularity for a Class of Quasilinear Elliptic Equations

The Lipschitz continuity of solutions to Dirichlet and Neumann problems for nonlinear elliptic equations, including the p-Laplace equation, is established under minimal integrability assumptions on the data and on the curvature of the boundary of the domain. The case of arbitrary bounded convex domains is also included. The results have new consequences even for the Laplacian.     

Local minimizers and rearrangements

We are concerned witha priori estimates for functionsu which locally minimize, in the topology ofL , functionals of the Calculus of Variations. Sharp pointwise upper bounds for the spherically symmetric rearrangement ofu are proved. Such result enables us to get conditions for the boundedness ofu and estimates for ess sup¦u¦.     

Functions of Bounded Variation¶and Rearrangements

The action of the symmetric rearrangement of functions of bounded variation is investigated. In particular, pointwise regularizing effects are established. Pólya-Szeg? type inequalities, with a characterization of equality cases, are also proved.     

Interpretation of the Mössbauer spectra of oxyhemoglobin in terms of a low lying excited state

It is shown how the temperature evolution of the Mössbauer spectra of oxyhemoglobin can be explained by means of an excited triplet state at about 150 K above the ground state. The agreement is particularly good for temperatures T ≤ 180 K.