Adriano Pisante

Bloch Bundles, Marzari-Vanderbilt Functional and Maximally Localized Wannier Functions

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

Uniaxial versus biaxial character of nematic equilibria in three dimensions

{d \leq 3}

Optimal Decay of Wannier functions in Chern and Quantum Hall Insulators

d≤3. The proof exploits ideas and methods from the theory of harmonic maps between Riemannian manifolds.

Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces

Abstract We consider weak solutions of the harmonic map flow between the three-dimensional unit ball B 3and the two-dimensional unit sphere, with as initial and boundary data. In this situation, we show the existence of infinitely many weak solutions. Indeed for any different from the origin we construct a weak solution whose singular set for all positive time is precisely Qand which satisfies a modified energy inequality.

A Global Compactness type result for Palais–Smale sequences in fractional Sobolev spaces

We study global minimizers of the Landau–de Gennes (LdG) energy functional for nematic liquid crystals, on arbitrary three-dimensional simply connected geometries with topologically non-trivial and physically relevant Dirichlet boundary conditions. Our results are specific to an asymptotic limit coined in terms of a dimensionless temperature and material-dependent parameter, t and some constraints on the material parameters, and we work in the

Two results on the equivariant Ginzburg–Landau vortex in arbitrary dimension

t\rightarrow \infty