Elliptic Obstacle Problems with Measurable Coefficients in Non-Smooth Domains

We establish a global weighted W ^{1, p} -regularity for solutions to variational inequalities and obstacle problems for divergence form elliptic systems with measurable coefficients in bounded non-smooth domains.     

Extremal Systems of Points and Numerical Integration on the Sphere

This paper considers extremal systems of points on the unit sphere S ^r⊂R ^r+1, related problems of numerical integration and geometrical properties of extremal systems. Extremal systems are systems of d _n =dim P _n points, where P _n is the space of spherical polynomials of degree at most n, which maximize the determinant of an interpolation matrix. Extremal systems for S ² of degrees up to 191 (36,864 points) provide well distributed points, and are found to yield interpolatory cubature rules with positive weights. We consider the worst case cubature error in a certain Hilbert space and its relation to a generalized discrepancy. We also consider geometrical properties such as the minimal geodesic distance between points and the mesh norm. The known theoretical properties fall well short of those suggested by the numerical experiments.

     

Strongly elliptic second-order systems with boundary conditions on a nonclosed Lipschitz surface

We consider boundary value problems and transmission problems for strongly elliptic second-order systems with boundary conditions on a compact nonclosed Lipschitz surface S with Lipschitz boundary. The main goal is to find conditions for the unique solvability of these problems in the spaces H ^s , the simplest L ₂-spaces of the Sobolev type, with the use of potential type operators on S. We also discuss, first, the regularity of solutions in somewhat more general Bessel potential spaces and Besov spaces and, second, the spectral properties of problems with spectral parameter in the transmission conditions on S, including the asymptotics of the eigenvalues.     

Local action functionals for randomly perturbed dynamical systems on long time intervals ∗

We prove exponential decays of probabilities of randomly perturbed dynamical systems in a d–dimensional Euclidean space R^d on time intervals which go to [0,∞] as the random fluctuation disappears. We also consider the exit problems when unperturbed dynamical systems are attracted to the inside of the domain under consideration     

Solution of a Certain Class of Problems of Optimal Control Theory in Domains of an Arbitrary Form

To solve optimal control problems for elliptic systems in domains that have an arbitrary form, we use a combination of the method of dummy domains and the grid method. We show that the proposed difference scheme has precision of order O(h ^1/2) with respect to the grid norm of W₂ ².     

Impulsive control systems with commutative vector fields

We consider variational problems with control laws given by systems of ordinary differential equations whose vector fields depend linearly on the time derivativeu=(u ¹,...,u ^m ) of the controlu=(u ¹,...,u ^m ). The presence of the derivativeu, which is motivated by recent applications in Lagrangian mechanics, causes an impulsive dynamics: at any jump of the control, one expects a jump of the state.The main assumption of this paper is the commutativity of the vector fields that multiply theu . This hypothesis allows us to associate our impulsive systems and the corresponding adjoint systems to suitable nonimpulsive control systems, to which standard techniques can be applied. In particular, we prove a maximum principle, which extends Pontryagin's maximum principle to impulsive commutative systems.     

Symmetry of solutions for quasimonotone second-order elliptic systems in ordered Banach spaces

We consider symmetry properties of solutions to nonlinear elliptic boundary value problems defined on bounded symmetric domains of \mathbb Rⁿ{\mathbb R^n} . The solutions take values in ordered Banach spaces E, e.g. E=\mathbb R^N{E=\mathbb R^N} ordered by a suitable cone. The nonlinearity is supposed to be quasimonotone increasing. By considering cones that are different from the standard cone of componentwise nonnegative elements we can prove symmetry of solutions to nonlinear elliptic systems which are not covered by previous results. We use the method of moving planes suitably adapted to cover the case of solutions of nonlinear elliptic problems with values in ordered Banach spaces.     

Zwei Klassen vollständiger Funktionensysteme zur Behandlung der Randwertaufgaben der Schwingungsgleichung ΔU + k2U = 0

In this paper we present new methods to solve the classical Dirichlet and Neumann problems for ΔU + k²U = 0. We prove that the solutions of this equation for a region S containing G restricted to G are dense in L²(?G). Introducing a basis in the space of solutions for S we find a complete orthogonal system in L²(?G) which can be used to solve the boundary value problems by means of approximation in the Hilbertspace norm. Regularity estimates lead to series expansions in G. The well-known basis systems obtained by separation of variables thus may be used for every regular region without the very special geometric restrictions. Another class of basis systems may be obtained in analogy to the Runge. theorems by considering types of singularity functions.     

Mixed problems in a Lipschitz domain for strongly elliptic second-order systems

We consider mixed problems for strongly elliptic second-order systems in a bounded domain with Lipschitz boundary in the space ℝ ⁿ . For such problems, equivalent equations on the boundary in the simplest L ₂-spaces H ^s of Sobolev type are derived, which permits one to represent the solutions via surface potentials. We prove a result on the regularity of solutions in the slightly more general spaces H _p ^s of Bessel potentials and Besov spaces B _p ^s . Problems with spectral parameter in the system or in the condition on a part of the boundary are considered, and the spectral properties of the corresponding operators, including the eigenvalue asymptotics, are discussed.     

An approximation procedure for solutions to Mayer problems with cost functional not of classC

We consider here control problems in the Mayer form, with a cost functional which is continuous, but not necessarily of classC ¹. The usual necessary conditions for such problems cannot be applied, since they require that the cost functional beC ¹. We describe a convergence procedure, based upon approximations to the cost functional, that will yield an optimal trajectory to such control systems. Criteria are given which justify that this approximation procedure will yield a trajectory and control, satisfying certain prescribed conditions, which minimize the above cost functional within a specified class of such pairs. Examples are presented.     

Correct and self-adjoint problems with cubic operators

In this paper, we present a simple method to prove the correctness and self-adjointness of operators B ³ corresponding to some boundary problems. We also give the unique solutions for these problems. The algorithm is easy to implement via computer algebra systems. In our examples, Derive and Mathematica were used. Bibliography: 10 titles.     

An Existence Theorem of Smooth Nonlocal Center Manifolds for Systems Close to a System with a Homoclinic Loop

Summary. In this paper we give a proof of the existence of smooth nonlocal center manifolds for systems close to a system with a homoclinic orbit to a saddle-type equilibrium point. Our proof is based on a consideration of some class of the boundary value problems (see Section 3). We obtain estimates for solutions of the boundary value problems that allow us to prove the theorem on the center manifolds at the C ¹ -assumptions for the smoothness of systems. Received June 4, 1997; final revision received April 24, 1998     

Some remarks on lengths of propositional proofs

We survey the best known lower bounds on symbols and lines in Frege and extended Frege proofs. We prove that in minimum length sequent calculus proofs, no formula is generated twice or used twice on any single branch of the proof. We prove that the number of distinct subformulas in a minimum length Frege proof is linearly bounded by the number of lines. Depthd Frege proofs ofm lines can be transformed into depthd proofs ofO(m ^d+1) symbols. We show that renaming Frege proof systems are p-equivalent to extended Frege systems. Some open problems in propositional proof length and in logical flow graphs are discussed. Supported in part by NSF grant DMS-9205181     

Einige Bemerkungen über die Regularität von stationären Punkten gewisser geometrischer Variationsintegrale

We discuss the regularity properties of weak solutions of certain degenerate elliptic systems arising from problems in geometry. For example we prove C¹-regularity of conformal m-harmonic maps from the m-ball into m-dimensional oriented submanifolds of R^m+1 as well as a regularity theorem for weak H-surfaces in R^m+1. Our arguments make strong use of a version of the isoperimetric inequality.     

Characteristic 2 approach to bivariate interpolation problems

We investigate bivariate Hermite interpolation problems in characteristic 2. Given a nonnegative integer t, we describe all the sub-linear systems generated by monomials, in which there is no curve passing through a general point with multiplicity at least 2 ^t . As an application, we show that a certain linear system of plane curves with ten base points is non-special.      

Dynamics groups of asynchronous cellular automata

We say that a finite asynchronous cellular automaton (or more generally, any sequential dynamical system) is π-independent if its set of periodic points are independent of the order that the local functions are applied. In this case, the local functions permute the periodic points, and these permutations generate the dynamics group. We have previously shown that exactly 104 of the possible 2²³=2562^{2^{3}}=256 cellular automaton rules are π-independent. In the article, we classify the periodic states of these systems and describe their dynamics groups, which are quotients of Coxeter groups. The dynamics groups provide information about permissible dynamics as a function of update sequence and, as such, connect discrete dynamical systems, group theory, and algebraic combinatorics in a new and interesting way. We conclude with a discussion of numerous open problems and directions for future research.     

A geometric view of Krylov subspace methods on singular systems

We give a geometric framework for analysing iterative methods on singular linear systems A x = b and apply them to Krylov subspace methods. The idea is to decompose the method into the ?(A) component and its orthogonal complement ?(A)^?, where ?(A) is the range of A. We apply the framework to GMRES, GMRES(k) and GCR(k), and derive conditions for convergence without breakdown for inconsistent and consistent singular systems. The approach also gives a geometric interpretation and different proofs of the conditions obtained by Brown and Walker for GMRES. We also give examples arising in the finite difference discretization of two‐point boundary value problems of an ordinary differential equation. Copyright © 2010 John Wiley & Sons, Ltd.     

Left-Definite Regular Hamiltonian Systems

Linear Hamiltonian systems allow us to generalize, as well as consider, self-adjoint problems of any even order. Such left-definite problems are interesting, not only because of the generalization, but also because of the new intricacies they expose, some of which have made it possible to go beyond fourth order scale problems. We explore the left definite Sobolev settings for such problems, which are in general subspaces determined by boundary conditions. We show that the Hamiltonian operator remains self-adjoint, and inherits the same resolvent and spectral resolution from its original L² space when set in the left-definite Sobolev space.     

Elementary divisors of induced transformations on tensors via the representation theory of sl 2

We study various stability type conditions on a matrix A related to the consistency of the Lyapunov equation AD+DA^t positive definite, where D is a positive diagonal matrix. Such problems arise in mathematical economics, in the study of time-invariant continuous-time systems and in the study of predator-prey systems. Using a theorem of the alternative, a characterization is given for all A satisfying the above equation. In addition, some necessary conditions for consistency and some related ideas are discussed. Finally, a method for constructing a solution D to the equation is given for matrices A satisfying certain conditions.     

Preconditioning a class of fourth order problems by operator splitting

We develop preconditioners for systems arising from finite element discretizations of parabolic problems which are fourth order in space. We consider boundary conditions which yield a natural splitting of the discretized fourth order operator into two (discrete) linear second order elliptic operators, and exploit this property in designing the preconditioners. The underlying idea is that efficient methods and software to solve second order problems with optimal computational effort are widely available. We propose symmetric and non-symmetric preconditioners, along with theory and numerical experiments. They both document crucial properties of the preconditioners as well as their practical performance. It is important to note that we neither need H ^s -regularity, s > 1, of the continuous problem nor quasi-uniform grids.