We obtain an improved Sobolev inequality in spaces involving Morrey norms. This refinement yields a direct proof of the existence of optimizers and the compactness up to symmetry of optimizing sequences for the usual Sobolev embedding. More generally, it allows to derive an alternative, more transparent proof of the profile decomposition in obtained in G,rard (ESAIM Control Optim Calc Var 3:213-233, 1998) using the abstract approach of dislocation spaces developed in Tintarev and Fieseler (Concentration compactness. Functional-analytic grounds and applications. Imperial College Press, London, 2007). We also analyze directly the local defect of compactness of the Sobolev embedding in terms of measures in the spirit of Lions (Rev Mat Iberoamericana 1:145-201, 1985, Rev Mat Iberoamericana 1:45-121, 1985). As a model application, we study the asymptotic limit of a family of subcritical problems, obtaining concentration results for the corresponding optimizers which are well known when is an integer (Rey in Manuscr Math 65:19-37, 1989, Han in Ann Inst Henri Poincar, Anal Non Lin,aire 8:159-174, 1991, Chou and Geng in Differ Integral Equ 13:921-940, 2000)

We consider a periodic Schrödinger operator and the composite Wannier functions corresponding to a relevant family of its Bloch bands, separated by a gap from the rest of the spectrum. We study the associated localization functional introduced in Marzari and Vanderbilt (Phys Rev B 56:12847-12865, 1997) and we prove some results about the existence and exponential localization of its minimizers, in dimension d le; 3. The proof exploits ideas and methods from the theory of harmonic maps between Riemannian manifolds. © 2013 Springer-Verlag Berlin Heidelberg

We investigate the localization properties of independent electrons in a periodic background, possibly including a periodic magnetic field, as e. g. in Chern insulators and in Quantum Hall systems. Since, generically, the spectrum of the Hamiltonian is absolutely continuous, localization is characterized by the decay, as |x| → ∞, of the composite (magnetic) Wannier functions associated to the Bloch bands below the Fermi energy, which is supposed to be in a spectral gap. We prove the validity of a localization dichotomy, in the following sense: either there exist exponentially localized composite Wannier functions, and correspondingly the system is in a trivial topological phase with vanishing Hall conductivity, or the decay of any composite Wannier function is such that the expectation value of the squared position operator, or equivalently of the Marzari-Vanderbilt localization functional, is +∞. In the latter case, the Bloch bundle is topologically non-trivial, and one expects a non-zero Hall conductivity.(2) Throughout this Section, we use Hartree atomic units, and moreover we reabsorb the reciprocal of the speed of light 1/c in the definition of the function A Γ .

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Abstract. We classify nonconstant entire local minimizers of the standard Ginzburg-Landau functional for maps in H 1 loc (R 3 ; R 3 ) satisfying a natural energy bound. Up to translations and rotations, such solutions of the Ginzburg-Landau system are given by an explicit solution equivariant under the action of the orthogonal group.