Semistable extremal ground states for nonlinear evolution equations in unbounded domains

In this paper we show that dissipative reaction-diffusion equations in unbounded domains posses extremal semistable ground states equilibria, which bound asymptotically the global dynamics. Uniqueness of such positive ground state and their approximation by extremal equilibria in bounded domains is also studied. The results are then applied to the important case of logistic equations.     

Irreducibility and p-adic monodromies on the Siegel moduli spaces

We generalize the surjectivity result of the p-adic monodromy for the ordinary locus of a Siegel moduli space by Faltings and Chai (independently by Ekedahl) to that for any p-rank stratum. We discuss irreducibility and connectedness of some p-rank strata of the moduli spaces with parahoric level structure. Finer results are obtained on the Siegel 3-fold with Iwahori level structure.     

On radial stochastic Loewner evolution in multiply connected domains

We discuss the extension of radial SLE to multiply connected planar domains. First, we extend Loewner's theory of slit mappings to multiply connected domains by establishing the radial Komatu-Loewner equation, and show that a simple curve from the boundary to the bulk is encoded by a motion on moduli space and a motion on the boundary of the domain. Then, we show that the vector-field describing the motion of the moduli is Lipschitz. We explain why this implies that “consistent,” conformally invariant random simple curves are described by multidimensional diffusions, where one component is a motion on the boundary, and the other component is a motion on moduli space. We argue what the exact form of this diffusion is (up to a single real parameter κ) in order to model boundaries of percolation clusters. Finally, we show that this moduli diffusion leads to random non-self-crossing curves satisfying the locality property if and only if κ=6.     

Observation of mechanical fracture and corresponding domain structure changes of polycrystalline PbTiO3 nanotubes

PbTiO₃ (PTO) nanotubes (NTs) were synthesized at various temperatures by gas phase reaction between PbO gas and anatase TiO₂ NTs and characterized by piezoresponse force microscopy (PFM). PTO ferroelectric phase was synthesized at as low as 400 °C as evidenced by PFM domain images and piezoresponse hysteresis loop measurement. Furthermore, the PTO NTs fabricated at above 500 °C underwent mechanical fracture through development of nanocracks due to grain growth, leading to ferroelectric domains with larger size and lower aspect ratio. (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A NEW ROPER-SUFFRIDGE EXTENSION OPERATOR ON BOUNDED COMPLETE REINHARDT DOMAINS

In this paper, we construct a new Roper-Suffridge extension operator Φr n,β1,,βn(f)(z) = F(z) = ((rf(z1/r)/z1)β1z1,(rf(z1/r)/z1)β2z2,...,(rf(z1/r)/z1)βnzn)',where f is a normalized locally biholomorphic function on the unit disc D, r = sup{|z1| : z =(z1, ···, zn) ∈Ω}, β1∈ [0, 1], 0 ≤βk≤β1, k = 2, ···, n, then we prove it can preserve the property of spirallikeness of type β, almost starlikeness of order α and starlikeness of orderα on bounded complete Reinhardt domain Ω, respectively.     

Holomorphic mappings preserving Minkowski functionals

We show that the equality _m1(f(x))=_m2(g(x))$m_1\left(f\left(x\right)\right)=m_2\left(g\left(x\right)\right)$ for x$x$ in a neighborhood of a point a$a$ remains valid for all x$x$ provided that f$f$ and g$g$ are open holomorphic maps, f(a)=g(a)=0$f\left(a\right)=g\left(a\right)=0$ and _m1,_m2$m_1,m_2$ are Minkowski functionals of bounded balanced domains. Moreover, a polynomial relation between f$f$ and g$g$ is obtained.     

Mechanical properties of amphiphilic urethane acrylate ionomer hydrogels having heterophasic gel structure

Amphiphilic urethane acryale hydrogels containing ionic groups (dimethylolpropionic acid) were prepared by varying the molecular weight of the soft segment (polyether type) and the type of diisocyanate, and their mechanical properties were examined. They showed heterophasic gel structure composed of ionic hard domains induced by aggregation of the ionic groups and polyether soft domains comprising the urethane acrylate network. This heterophasic structure could be confirmed by dynamic mechanical analysis (DMA) and by wide-angle X-ray scattering analysis (WAXS); the crystallinity detected by WAXS and the transition peak of the ionic hard domains detected by DMA strongly suggested that there were ionic aggregates. These ionic aggregates acted as reinforcing fillers in the network, which eventually enhanced the tensile strength of the hydrogels. Above all, the tensile properties of the hydrogels were of interest in that the trends of the stress-strain curves were consistent with the rubbery ones. It is believed that the higher purity of the polyether soft domains resulted from the heterophasic gel structure imparting further elastomeric properties on the network. Received: 31 July 1998 Accepted in revised form: 15 October 1998     

The Neumann problem and Helmholtz decomposition in convex domains

We show that the Neumann problem for Laplace's equation in a convex domain Ω with boundary data in Lp(∂Ω) is uniquely solvable for 1<p<∞. As a consequence, we obtain the Helmholtz decomposition of vector fields in Lp(Ω,Rd).