Metastability and Dynamics of Discrete Topological Singularities in Two Dimensions: A Γ-Convergence Approach

This paper aims at building a variational approach to the dynamics of discrete topological singularities in two dimensions, based on Γ-convergence. We consider discrete systems, described by scalar functions defined on a square lattice and governed by periodic interaction potentials. Our main motivation comes from XY spin systems, described by the phase parameter, and screw dislocations, described by the displacement function. For these systems, we introduce a discrete notion of vorticity. As the lattice spacing tends to zero we derive the first order Γ-limit of the free energy which is referred to as renormalized energy and describes the interaction of vortices. As a byproduct of this analysis, we show that such systems exhibit increasingly many metastable configurations of singularities. Therefore, we propose a variational approach to the depinning and dynamics of discrete vortices, based on minimizing movements. We show that, letting first the lattice spacing and then the time step of the minimizing movements tend to zero, the vortices move according with the gradient flow of the renormalized energy, as in the continuous Ginzburg–Landau framework.     

Ligand-controlled regioselectivity in the hydrothiolation of alkynes by rhodium N-heterocyclic carbene catalysts

Rh-N-heterocyclic carbene compounds [Rh(μ-Cl)(IPr)(η(2)-olefin)](2) and RhCl(IPr)(py)(η(2)-olefin) (IPr = 1,3-bis(2,6-diisopropylphenyl)imidazol-2-carbene, py = pyridine, olefin = cyclooctene or ethylene) are highly active catalysts for alkyne hydrothiolation under mild conditions. A regioselectivity switch from linear to 1-substituted vinyl sulfides was observed when mononuclear RhCl(IPr)(py)(η(2)-olefin) catalysts were used instead of dinuclear precursors. A complex interplay between electronic and steric effects exerted by IPr, pyridine, and hydride ligands accounts for the observed regioselectivity. Both IPr and pyridine ligands stabilize formation of square-pyramidal thiolate-hydride active species in which the encumbered and powerful electron-donor IPr ligand directs coordination of pyridine trans to it, consequently blocking access of the incoming alkyne in this position. Simultaneously, the higher trans director hydride ligand paves the way to a cis thiolate-alkyne disposition, favoring formation of 2,2-disubstituted metal-alkenyl species and subsequently the Markovnikov vinyl sulfides via alkenyl-hydride reductive elimination. DFT calculations support a plausible reaction pathway where migratory insertion of the alkyne into the rhodium-thiolate bond is the rate-determining step.     

Gradient integrability and rigidity results for two-phase conductivities in two dimensions

This paper deals with higher gradient integrability for σ-harmonic functions u with discontinuous coefficients σ  , i.e. weak solutions of div(σ∇u)=0$\mathrm{div}(\sigma \mathrm{\nabla }u)=0$ in dimension two. When σ is assumed to be symmetric, then the optimal integrability exponent of the gradient field is known thanks to the work of Astala and Leonetti and Nesi. When only the ellipticity is fixed and σ is otherwise unconstrained, the optimal exponent is established, in the strongest possible way of the existence of so-called exact solutions, via the exhibition of optimal microgeometries.     

Semi-linear Wave Equations with Effective Damping

The authors study the Cauchy problem for the semi-linear damped wave equation $$u_{tt} - \Delta u + b\left( t \right)u_t = f\left( u \right), u\left( {0,x} \right) = u_0 \left( x \right), u_t \left( {0,x} \right) = u_1 \left( x \right)$$ in any space dimension n ≥ 1. It is assumed that the time-dependent damping term b(t) > 0 is effective, and in particular tb(t) → ∞ as t → ∞. The global existence of small energy data solutions for |f(u)| ≈ |u| ^p in the supercritical case of $p > \tfrac{2} {n}$ and $p \leqslant \tfrac{n} {{n - 2}}$ for n ≥ 3 is proved.     

A minimax theorem in the presence of unbounded Palais-Smale sequences

In a reflexive Banach space we consider a family of functionals that may admit unbounded Palais-Smale sequences. Under some structural conditions we provide a suitable Deformation Lemma that is obtained by modifying the classical pseudo-gradient flow. This leads to a variant of the minimax principle in the Lusternik-Schnirelman Theory.     

Two-dimensional nanoscale self-assembly on a gold surface by spinodal decomposition

The structure formation upon spinodal decomposition of a two-dimensional model system, a Au adatom gas on a Au(111) surface, was observed in situ by scanning tunneling microscopy (STM). A thermodynamically unstable state was prepared by applying microsecond voltage pulses to the STM tip in an electrochemical system, causing the random dissolution of Au atoms from the uppermost monolayer. Interconnected, labyrinthine island patterns were formed at Au coverages between 0.4 and 0.9 monolayer with dominating length scales lambda(m) of the order of a few nanometers.