On degenerate backward SPDEs in bounded domains under non-local conditions

ABSTRACT

The paper studies backward stochastic partial differential equations (BSPDEs) of parabolic type in bounded domains in the setting where the coercivity condition is not necessary satisfied and under special non-local in time and space boundary conditions replacing the standard Cauchy condition. These conditions connect the terminal value of the solution with a functional over the entire past solution. Uniqueness, solvability, and regularity results are obtained. As an example of applications, it is shown that degenerate BSPDEs with non-local boundary conditions arise naturally in modelling of portfolio selection problems, including models where dividend payoffs and management fees are taken into account.     

Bernstein <Emphasis Type="Italic">n</Emphasis>-widths of Besov embeddings on Lipschitz domains

In this paper, using an equivalent characterization of the Besov space by its wavelet coefficients and the discretization technique due to Maiorov, we determine the asymptotic degree of the Bernstein n-widths of the compact embeddings Bq0s＋t（Lp0（Ω））→Bq1s（Lp1（Ω））, t〉max{d（1/p0-1/p1）, 0}, 1 ≤ p0, p1, q0, q1 ≤∞,where Bq0s＋t（Lp0（Ω）） is a Besov space defined on the bounded Lipschitz domain Ω ？ Rd. The results we obtained here are just dual to the known results of Kolmogorov widths on the related classes of functions.     

Existence of solutions to second-order nonlinear discrete elliptic equations

In this paper, we consider a boundary value problem (BVP) for second-order nonlinear partial difference equations on finite lattice domains. Some conditions are established that ensure existence and uniqueness of solutions to the BVP under consideration.     

二维自治g-Navier-Stokes系统的全局渐近行为研究

研究了有界区域上二维自治g-Navier-Stokes系统的双全局吸引子,利用非紧性测度方法,给出了一种验证其存在性的新方法.得出二维自治g-Navier-Stokes方程在有界区域上有一个非空、紧可逆H_g一V_g全局吸引子这一结论.     

Renormalization phenomena in Ba-diluted ferroelastic lead phosphate, (Pb1-x Ba x )3(PO4)2

Abstract

In the improper ferroelastic palmierite-type lead phosphate order parameter coupling with a defect induced conjugate field leads to the renormalization of the two different critical temperatures of three order parameter components {Q ₃} and {Q ₁ Q ₂}. The influence of the lead dilution by barium on the ferroelastic domain pattern, the critical temperature of the ferroelastic transformation R m–C2/c and the development of the intermediate regime in (Pb_1–x Ba _x )₃(PO₄)₂ is studied using optical birefringence measurements, Raman and infrared spectroscopy. At the ferroelastic transition temperature T _c the orientational contribution of the three-states Potts model becomes critical. T _c is reduced from 453 K (x = 0) to zero K at x? 0.12. Modifications of the shape of zigzag needle domains as well as the angle between the monoclinic binary axis and the W walls along [031] with temperature and increasing Ba-content are reported. Above the ferroelastic transition point the component Q ₃, which corresponds to the displacive part in the Gibbs free energy, leads to dynamic short-range monoclinic deformation in the trigonal matrix. The temperature where Q ₃ shows critical behaviour is renormalized to 720 K (x = 0.12) as compared with 563 K in pure lead phosphate. For x>0.13 no monoclinic precursor clusters were found.     

Ferroelastic detwinning of the high-temperature superconductor YBa2Cu3O7-δ

Crystals of YBa₂Cu₃O_7-δ (YBC) were prepared using a flux technique. The as-grown crystals were typically on the order of 200 μm × 200 μm × 50 μm, where the 50 μm dimension is in the c-direction. Ferroelastic behavior was observed by applying a uniaxial stress along the [100]_tet-direction using a specially-designed fixture which was mounted to a programmable hot/cold stage of a polarizing reflected-light microscope. Domain wall motion was observed for temperatures greater than 200°C, and complete poling was accomplished at stresses on the order of 50–100 MPa at temperatures of 350–400°C. An ionic diffusion-activated mechanism is suggested.     

On the Uniform Poincaré Inequality

We give a proof of the Poincaré inequality in W ^{1, p} (Ω) with a constant that is independent of Ω ? , where  is a set of uniformly bounded and uniformly Lipschitz domains in ? ⁿ . As a byproduct, we obtain the following: The first non vanishing eigenvalues λ₂(Ω) of the standard Neumann (variational) boundary value problem on Ω for the Laplace operator are bounded below by a positive constant if the domains Ω vary and remain uniformly bounded and uniformly Lipschitz regular.     

Behavior of Solutions to the Dirichlet Problem for Elliptic Systems in Convex Domains

We consider the Dirichlet problem for strongly elliptic systems of order 2m in convex domains. Under a positivity assumption on the Poisson kernel it is proved that the weak solution has bounded derivatives up to order m provided the outward unit normal has no big jumps on the boundary. In the case of second order symmetric systems in plane convex domains the boundedness of the first derivatives is proved without the assumption on the normal.     

Spatial and Spatiotemporal Karhunen-Loève-Type Representations on Fractal Domains

Abstract

We study the spectral properties of spatial and spatiotemporal Gaussian random fields defined as the solutions to stochastic elliptic, parabolic, and hyperbolic fractional pseudodifferential equations on compact fractal domains. The fractal dimension of the domain modifies the asymptotic properties of the eigenvalues that define the pure point spectra of the covariance functions of the solutions and their Karhunen-Loève-type expansions. The eigenfunction systems involved constitute orthogonal bases of the corresponding trace spaces on fractal sets. The Hölder exponent of the sample paths of the random fields is computed in terms of the fractional order of mean-quadratic variation on their increments. Such an exponent also depends on the Hausdorff dimension of the domain.