Generalized BSDEs and nonlinear Neumann boundary value problems

Summary. We study a new class of backward stochastic differential equations, which involves the integral with respect to a continuous increasing process. This allows us to give a probabilistic formula for solutions of semilinear partial differential equations with Neumann boundary condition, where the boundary condition itself is nonlinear. We consider both parabolic and elliptic equations. Received: 27 September 1996 / In revised form: 1 December 1997     

Maximum principles, uniqueness and existence for harmonic maps with potential and Landau-Lifshitz equations

In this paper, we consider the harmonic maps with potential from compact Riemannian manifold with boundary into a convex ball in any Riemannian manifold. We will establish some general properties such as the maximum principles, uniqueness and existence for these maps, and as an application of them, we derive existence and uniqueness result for the Dirichlet problem of the Landau-Lifshitz equations. Received: December 10, 1997 / Accepted: June 29, 1998     

Nonlinear degenerate elliptic equations and axially symmetric problems

We consider semilinear elliptic equations with a principal part degenerating on a boundary hyperplane. Weak existence, uniqueness and regularity of solutions are established by variational methods and by reduction to uniformly elliptic equations. An important application arises in the mathematical treatment of the rotating star problem in general relativity, where the axial symmetry admits the reduction of one of the Einstein equations to a problem of the above form on a meridian half plane. Received February 12, 1997 / Accepted May 15, 1997     

On the Complete Integrability of some Lax Equations on a Periodic Lattice

We consider some Lax equations on a periodic lattice with sites under which the monodromy matrix evolves according to the Toda flows. To establish their integrability (in the sense of Liouville) on generic symplectic leaves of the underlying Poisson structure, we construct the action-angle variables explicitly. The action variables are invariants of certain group actions. In particular, one collection of these invariants is associated with a spectral curve and the linearization of the associated Hamilton equations involves interesting new feature. We also prove the injectivity of the linearization map into real variables and solve the Hamilton equations generated by the invariants via factorization problems.

     

Differential equations with 2-term recursion

In this paper we consider linear differential equations with a recursion consisting of two terms. We consider these equations in positive characteristics and in characteristic zero. We will find a new proof for the Grothendieck conjecture for these equations. Supported by a grant from the DFG.     

Scaling Methods and Approximative Equations for Homogeneous Reaction—Diffusion Systems and Applications to Epidemics

Summary. We consider a reaction-diffusion equation that is homogeneous of degree one. This homogeneity is a symmetry. The dynamics is factorized into trivial evolution due to symmetry and nontrivial behavior by a projection to an appropriate hypermanifold. The resulting evolution equations are rather complex. We examine the bifurcation behavior of a stationary point of the projected system. For these purposes we develop techniques for dimension reduction similar to the Ginzburg-Landau (GL) approximation, the modulation equations. Since we are not in the classical GL situation, the remaining approximative equations have a quadratic nonlinearity and the amplitude does not scale with ε but with ε ² where ε ² denotes the bifurcation parameter. Moreover, the symmetry requires that not only one but two equations are necessary to describe the behavior of the system. We investigate traveling fronts for the modulation equations. This result is used to analyze an epidemic model. Received April 9, 1996; second revision received January 3, 1997; final revision received October 7, 1997; accepted January 19, 1998     

Preconditioning in H and applications

We consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator I. The natural setting for such problems is in the Hilbert space H and the variational formulation is based on the inner product in H. We show how to construct preconditioners for these equations using both domain decomposition and multigrid techniques. These preconditioners are shown to be spectrally equivalent to the inverse of the operator. As a consequence, they may be used to precondition iterative methods so that any given error reduction may be achieved in a finite number of iterations, with the number independent of the mesh discretization. We describe applications of these results to the efficient solution of mixed and least squares finite element approximations of elliptic boundary value problems.

     

A Scaling Limit of the Becker-Döring Equations in the Regime of Small Excess Density

The Becker-Doring equations serve as a model for the nucleation of a new thermodynamic phase in a first-order phase transformation. This corresponds to the case when the total density of monomers exceeds a critical value and the excess density is contained in larger and larger clusters as time proceeds. It has been derived in Penrose [J. Stat. Phys. 89:1/2 (1997), 305-320] and Niethammer [J. Nonlin. Sci. 13:1 (2003), 115-155] that the evolution of these large clusters can on a certain large time scale be described by a nonlocal transport equation coupled with the constraint that the total volume of new phase is conserved. For specific coefficients this equation is well known as a classical mean-field model for coarsening. In the present paper we consider the regime of small excess density on a large time scale, but not as large as in Penrose (1997) or Niethammer (2003). We show rigorously that the leading order dynamics are governed by another variant of the classical mean-field model in which total mass is preserved.     

Derivation and classification of Vlasov-type and magnetohydrodynamics equations: Lagrange identity and Godunov’s form

We describe the derivation of the Vlasov-Maxwell equations from the Lagrangian of classical electrodynamics, from which magnetohydrodynamic-type equations are in turn derived. We consider both the relativistic and nonrelativistic cases: with zero temperature as the exact consequence of the Vlasov-Maxwell equations and with nonzero temperature as a zeroth-order approximation of the Maxwell-Chapman-Enskog method. We obtain the Lagrangian identities and their generalizations for these cases and compare them.     

On one generalization of the averaging method

We consider one case where it is possible to establish sufficient conditions for the convergence and analyticity of matrix series used for the construction of a system of moment equations. Kiev National Economic University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 7, pp. 906–911, July, 1997.     

Construction of locally conservative fluxes for the SUPG method

We consider the construction of locally conservative fluxes by means of a simple postprocessing technique obtained from the finite element solutions of advection diffusion equations. It is known that a naive calculation of fluxes from these solutions yields nonconservative fluxes. We consider two finite element methods: the usual continuous Galerkin finite element method for solving nondominating advection diffusion equations and the streamline upwind/Petrov‐Galerkin method for solving advection dominated problems. We then describe the postprocessing technique for constructing conservative fluxes from the numerical solutions of the general variational formulation. The postprocessing technique requires solving an auxiliary Neumann boundary value problem on each element independently and it produces a locally conservative flux on a vertex centered dual mesh relative to the finite element mesh. We provide a convergence analysis for the postprocessing technique. Performance of the technique and the convergence behavior are demonstrated through numerical examples including a set of test problems for advection diffusion equations, advection dominated equations, and drift‐diffusion equations. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1971–1994, 2015     

Diffusion processes on graphs: stochastic differential equations,large deviation principle

Ito's rule is established for the diffusion processes on the graphs. We also consider a family of diffusions processes with small noise on a graph. Large deviation principle is proved for these diffusion processes and their local times at the vertices. Received: 12 February 1997 / Revised version: 3 March 1999     

Multiplicity results for the two-vortex Chern-Simons Higgs model on the two-sphere

We consider a Ginzburg-Landau type functional on S ² with a 6 ^th order potential and the corresponding selfduality equations. We study the limiting behavior in the two vortex case when a coupling parameter tends to zero. This two vortex case is a limiting case for the Moser inequality, and we correspondingly detect a rich and varied asymptotic behavior depending on the position of the vortices. We exploit analogies with the Nirenberg problem for the prescribed Gauss curvature equation on S ². Received: December 3, 1997     

On the Equivalence of Solutions for a Class of Stochastic Evolution Equations in a Banach Space

We study a class of stochastic evolution equations in a Banach space E driven by cylindrical Wiener process. Three different analytical concepts of solutions: generalised strong, weak and mild are defined and the conditions under which they are equivalent are given. We apply this result to prove existence, uniqueness and continuity of weak solutions to stochastic delay evolution equations. We also consider two examples of these equations in non-reflexive Banach spaces: a stochastic transport equation with delay and a stochastic delay McKendrick equation.     

Carleman problem for two pairs of functions in a ring

We consider a particular case of the matrix Carleman problem for two pairs of functions in a ring and find a constructive solution of this problem. In addition, we propose an algorithm for the construction of solutions for two infinite systems of smooth transition and for a system of two singular equations of special type. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 5, pp. 662–671, May, 1997     

Nondivergent elliptic equations on manifolds with nonnegative curvature

We consider a class of second-order linear elliptic operators, intrinsically defined on Riemannian manifolds, that correspond to nondivergent operators in Euclidean space. Under the assumption that the sectional curvature is nonnegative, we prove a global Krylov-Safonov Harnack inequality and, as a consequence, a Liouville theorem for solutions of such equations. From the Harnack inequality, we obtain Alexandroff-Bakelman-Pucci estimates and maximum principles for subsolutions. © 1997 John Wiley & Sons, Inc.     

Extension of a piezoceramic bimorph with a crack crossing the interface

We consider the boundary-value problem of electroelasticity for a composite plate weakened by a crack crossing the line joining the media. The initial boundary-value problem, is reduced to a mixed system of singular integral and algebraic equations. We present the calculation results characterizing the variation in the stress intensity factors as a function of the opening angle of a segmented crack for different types of loading.Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 4, pp. 482–488, July–August 1997.     

Kostin-type criterion for abstract linear differential equations of arbitrary order in banach spaces

We consider linear differential equations with operator coefficients in a Banach space. We construct necessary and sufficient conditions for the well-posedness of the Cauchy problem for these equations of arbitrary order that are analogous to the Kostin conditions for incomplete equations of the second order.     

Chaos in singular systems

In this paper we consider singular systems of differential equations and we show that, under right conditions, the Poincaré map associated to those systems, and not just a suitable iterate, behaves chaotically. We use the notion of exponential dichotomies to prove the existence of a transverse homoclinic orbit of our system and after use the shadow lemma to show that the Poincaré map associated to its topologically conjugate to the Bernouilli shift on a set of two symbols. Entrata in Redazione il 3 aprile 1997 e, in versione riveduta, il 30 ottobre 1997.     

Multidimensional characteristic Galerkin methods for hyperbolic systems

We consider characteristic Galerkin methods for the solution of hyperbolic systems of partial differential equations of first order. A new recipe for the construction of approximate evolution operators is given in order to derive consistent methods. With the help of semigroup theory we derive error estimates for classes of characteristic Galerkin methods. The theory is applied to the wave equation and also to the Euler equations of gas dynamics. In the latter case one can show that Fey's genuinely multidimensional method can be reinterpreted as a characteristic Galerkin method. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.