A Comparison of Splittings and Integral Equation Solvers for a Nonseparable Elliptic Equation

Iterative numerical schemes for solving the electrostatic partial differential equation with variable conductivity, using fast and high-order accurate direct methods for preconditioning, are compared. Two integral method schemes of this type, based on previously suggested splittings of the equation, are proposed, analyzed, and implemented. The schemes are tested for large problems on a square. Particular emphasis is paid to convergence as a function of geometric complexity in the conductivity. Differences in performance of the schemes are predicted and observed in a striking manner. AMS subject classification (2000) 31A10, 35C15, 65R20.Received May 2004. Accepted September 2004. Communicated by Anders Szepessy.Johan Helsing: This work was supported by the Swedish Science Research Council under contract 621-2001-2799.     

Optimal Control Problem Associated with Jump Processes

An optimal portfolio/control problem is considered for a two-dimen\-sional model in finance. A pair consisting of the wealth process and cumulutative consumption process driven by a geometric Lévy process is controlled by adapted processes. The value function appears and turns out to be a viscosity solution to some integro-differential equation, by using the Bellman principle.     

Entire solutions of semilinear elliptic equations in and a conjecture of De Giorgi

In 1978 De Giorgi formulated the following conjecture. Let be a solution of in all of such that and 0$"> in . Is it true that all level sets of are hyperplanes, at least if ? Equivalently, does depend only on one variable? When , this conjecture was proved in 1997 by N. Ghoussoub and C. Gui. In the present paper we prove it for . The question, however, remains open for . The results for and 3 apply also to the equation for a large class of nonlinearities .

     

Numerical experiments with several variant WENO schemes for the Euler equations

Numerical experiments with several variants of the original weighted essentially non‐oscillatory (WENO) schemes (J. Comput. Phys. 1996; 126 :202–228) including anti‐diffusive flux corrections, the mapped WENO scheme, and modified smoothness indicator are tested for the Euler equations. The TVD Runge–Kutta explicit time‐integrating scheme is adopted for unsteady flow computations and lower–upper symmetric‐Gauss–Seidel (LU‐SGS) implicit method is employed for the computation of steady‐state solutions. A numerical flux of the variant WENO scheme in flux limiter form is presented, which consists of first‐order and high‐order fluxes and allows for a more flexible choice of low‐order schemes. Computations of unsteady oblique shock wave diffraction over a wedge and steady transonic flows over NACA 0012 and RAE 2822 airfoils are presented to test and compare the methods. Various aspects of the variant WENO methods including contact discontinuity sharpening and steady‐state convergence rate are examined. By using the WENO scheme with anti‐diffusive flux corrections, the present solutions indicate that good convergence rate can be achieved and high‐order accuracy is maintained and contact discontinuities are sharpened markedly as compared with the original WENO schemes on the same meshes. Copyright © 2008 John Wiley & Sons, Ltd.     

Gradient estimates in Orlicz space for nonlinear elliptic equations

In this paper we generalize gradient estimates in Lp space to Orlicz space for weak solutions of elliptic equations of p-Laplacian type with small BMO coefficients in δ-Reifenberg flat domains. Our results improve the known results for such equations using a harmonic analysis-free technique.     

Hartogs Phenomena and Antisyzygies for Systems of Differential Equations

We introduce the notion of antisyzygies, which studies the inverse problem of finding a system of PDEs, given compatibility conditions. The system obtained possesses the property of removability of compact singularities. We write explicit computations in the cases of the Cauchy-Fueter system and Maxwell’s system for electromagnetism, and we conclude with a study of systems of non-maximal rank.      

On the Elliptic Equation △u ＋ K（x）e^2u -0 with K（x） Positive Somewhere

This paper considers the existence problem of an elliptic equation.which is equivalent to the prescribing conformal Gaussian curvature problem on R2.An existence result is pried.In particular,K(x)is allowed to be unbounded above.     

Pd-catalyzed intramolecular direct arylations at high temperature

Pd-catalyzed cyclodehydrohalogenation involving oxidative addition of aromatic C-Br or activated azaheteroaromatic C-Cl bonds and C(sp²)-H activation have been investigated at high reaction temperatures (180-200 °C). This allowed the fast (10-30 min) synthesis of a variety of azaheteroaromatic ring systems (dibenzo[f,h]phthalazine, dibenzo[f,h]cinnoline, benzofuro[2,3-d]pyridazine, 5H-pyridazino[4,5-b]indole, 7H-indolo[2,3-c]quinoline and 5H-δ-carboline) in moderate to good yields.     

A new class of three-variable orthogonal polynomials and their recurrences relations

A new class of three-variable orthogonai polynomials,defined as eigenfunctions of a second order PDE operator,is studied.These polynomials are orthogonal over a curved tetrahedron region, which can be seen as a mapping from a traditional tetrahedron,and can be taken as an extension of the 2-D Steiner domain.The polynomials can be viewed as Jacobi polynomials on such a domain.Three- term relations are derived explicitly.The number of the individual terms,involved in the recurrences relations,are shown to be independent on the total degree of the polynomials.The numbers now are determined to be five and seven,with respect to two conjugate variables z,(?) and a real variable r, respectively.Three examples are discussed in details,which can be regarded as the analogues of the Chebyshev polynomials of the first and the second kinds,and Legendre polynomials.     

Global existence of strong solutions to the one-dimensional full model for phase transitions in thermoviscoelastic materials

This paper is devoted to the analysis of a one-dimensional model for phase transition phenomena in thermoviscoelastic materials. The corresponding parabolic-hyperbolic PDE system features a strongly nonlinear internal energy balance equation, governing the evolution of the absolute temperature ϑ, an evolution equation for the phase change parameter χ, including constraints on the phase variable, and a hyperbolic stress-strain relation for the displacement variable u. The main novelty of the model is that the equations for χ and u are coupled in such a way as to take into account the fact that the properties of the viscous and of the elastic parts influence the phase transition phenomenon in different ways. However, this brings about an elliptic degeneracy in the equation for u which needs to be carefully handled. First, we prove a global well-posedness result for the related initial-boundary value problem. Secondly, we address the long-time behavior of the solutions in a simplified situation. We prove that the ω-limit set of the solution trajectories is nonempty, connected and compact in a suitable topology, and that its elements solve the steady state system associated with the evolution problem. Dedicated to Jürgen Sprekels on the occasion of his 60th birthday