The Gaussian free field and Hadamard's variational formula

We relate the Gaussian free field on a planar domain to the variational formula of Hadamard which explains the change of the Green function under a perturbation of the domain. This is accomplished by means of a natural integral operator related to Hadamard's formula.Comment: 9 page

Weak compactness and essential norms of integration operators

Let $g$ be an analytic function on the unit disc and consider the integration operator of the form $T_g f(z) = \int_0^z fg'\,d\zeta$. We show that on the spaces $H^1$ and $BMOA$ the operator $T_g$ is weakly compact if and only if it is compact. In the case of $BMOA$ this answers a question of Siskakis and Zhao. More generally, we estimate the essential and weak essential norms of $T_g$ on $H^p$ and $BMOA$

Molecular ordering of precursor films during spreading of tiny liquid droplets

In this work we address a novel feature of spreading dynamics of tiny liquid droplets on solid surfaces, namely the case where the ends of the molecules feel different interactions to the surface. We consider a simple model of dimers and short chain--like molecules which cannot form chemical bonds with the surface. We study the spreading dynamics by Molecular Dynamics techniques. In particular, we examine the microscopic structure of the time--dependent precursor film and find that in some cases it can exhibit a high degree of local order. This order persists even for flexible chains. Our results suggest the possibility of extracting information about molecular interactions from the structure of the precursor film.Comment: 4 pages, revtex, no figures, complete file available from ftp://rock.helsinki.fi/pub/preprints/tft/ or at http://www.physics.helsinki.fi/tft/tft_preprints.html (to appear in Phys. Rev. E Rapid Comm.

Dynamics of Spreading of Small Droplets of Chainlike Molecules on Surfaces

Dynamics of spreading of small droplets on surfaces has been studied by the molecular dynamics method. Simulations have been performed for mixtures of solvent and dimer, and solvent and tetramer droplets. For solvent particles and dimers, layering occurs leading to stepped droplet shapes. For tetramers such shapes occur for relatively deep and strong surface potentials only. For wider and more shallow potentials, more rapid spreading and rounded droplet shapes occur. These results are in accordance with experimental data on small non - volatile polymer droplets. PACS numbers: 68.10Gw, 05.70.Ln, 61.20.Ja, 68.45GdComment: to appear in Europhys. Letters (1994), Latex, 12 page

Grazing-angle scattering of electromagnetic waves in gratings with varying mean parameters: grating eigenmodes

A highly unusual pattern of strong multiple resonances for bulk electromagnetic waves is predicted and analysed numerically in thick periodic holographic gratings in a slab with the mean permittivity that is larger than that of the surrounding media. This pattern is shown to exist in the geometry of grazing-angle scattering (GAS), that is when the scattered wave (+1 diffracted order) in the slab propagates almost parallel to the slab (grating) boundaries. The predicted resonances are demonstrated to be unrelated to resonant generation of the conventional guided modes of the slab. Their physical explanation is associated with resonant generation of a completely new type of eigenmodes in a thick slab with a periodic grating. These new slab eigenmodes are generically related to the grating; they do not exist if the grating amplitude is zero. The field structure of these eigenmodes and their dependence on structural and wave parameters is analysed. The results are extended to the case of GAS of guided modes in a slab with a periodic groove array of small corrugation amplitude and small variations in the mean thickness of the slab at the array boundaries.Comment: 16 pages, 6 figure

Wigner molecules in polygonal quantum dots: A density functional study

We investigate the properties of many-electron systems in two-dimensional polygonal (triangle, square, pentagon, hexagon) potential wells by using the density functional theory. The development of the ground state electronic structure as a function of the dot size is of particular interest. First we show that in the case of two electrons, the Wigner molecule formation agrees with the previous exact diagonalization studies. Then we present in detail how the spin symmetry breaks in polygonal geometries as the spin density functional theory is applied. In several cases with more than two electrons, we find a transition to the crystallized state, yielding coincidence with the number of density maxima and the electron number. We show that this transition density, which agrees reasonably well with previous estimations, is rather insensitive to both the shape of the dot and the electron number.Comment: 8 pages, 11 figure