Nonclassical Solutions of Fully Nonlinear Elliptic Equations

We prove the existence of non-smooth solutions to fully non-linear elliptic equations. Received: June 2006, Accepted: January 2007     

Representation of Contractive Solutions of a Class of Algebraic Riccati Equations as Characteristic Functions of Maximal Dissipative Operators

Let

Projection-Iterative Methods for a Class of Difference Equations

We develop a theoretical framework for projection-iterative methods to solve operator equations of the form Au + Bu = f, where A is a Toeplitz operator in a Banach space , B is considered as a perturbation (of general form) of A, and f is a given element in this space. The methods are adopted for application to general situations, in particular, to the equations in which A need not be a Fredholm operator. The idea to involve iteration procedures and the technique which we apply allow to obtain conditions on perturbations for convergence and effective error estimates in terms of some weighted spaces (without any restrictions on the norms for perturbations). Based on established evaluations we derive further information about decaying properties of the solutions. The obtained results are illustrated by considering concrete classes of equations as, for instance, equations corresponding to Jacobi type operators.      

Critical Nonlinear Nonlocal Equations on a Half-Line

We study nonlinear nonlocal equations on a half-line in the critical case

Blow-up Solutions for Mixed Nonlinear Schrodinger Equations

This paper is concerned with the initial boundary-value problem for the following nonlinear evolution equation: Фt=iαФxx βФ^2Ф^-x γ|Ф|^2Фx ig(|Ф|^2)Ф Under certain conditions on the initial data and the function g(s),we study the existence and nonexistence of global solution for this equation.The blow-up solution and the blow-up time are also investigated.     

Homological Dimension and Critical Exponent of Kleinian Groups

We prove an inequality between the relative homological dimension of a Kleinian group and its critical exponent. As an application of this result we show that for a geometrically finite Kleinian group Γ, if the topological dimension of the limit set of Γ equals its Hausdorff dimension, then the limit set is a round sphere. Received: March 2007, Revision: October 2007, Accepted: October 2007     

Non-global existence for an Allen-Cahn-Gurtin equation with logarithmic free energy

We consider a simple Allen-Cahn-Gurtin model with a logarithmic potential. We show that strong solutions exist locally in time, but not globally in general. This is due to a lack of maximum principle.      

Regular Problems for Semilinear Hyperbolic Type Equations

In this paper we establish the wellposedness and regularity properties of solutions of Cauchy problems for semilinear hyperbolic equations of second order with unbounded principal operators. An example illustrating how our results apply is given.      

A sharp estimate and change on the dimension of the attractor for singular semilinear parabolic equations

We consider the semilinear reaction diffusion equation

Equations in finite fields with restricted solution sets. II (Algebraic equations)

Generalizing earlier results, it is shown that if are “large” subsets of a finite field F _q , then the equations a + b = cd, resp. ab + 1 = cd can be solved with . Other algebraic equations with solutions restricted to “large” subsets of F _q are also studied. The proofs are based on character sum estimates proved in Part I of the paper. Research partially supported by the Hungarian National Foundation for Scientific Research, Grants No. T 043623, T 043631 and T 049693.     

Nonlinear electroelastostatics: a variational framework

Different formulations of the constitutive laws and governing equations for nonlinear electroelastic solids are reviewed and two new variational principles for electroelastostatics are introduced. One is based on use of the electrostatic scalar potential and one on the vector potential, combined with the deformation function. In each case Lagrangian forms of the electric variables are used. Their connections with several formulations of nonlinear electroelasticity in the literature are established and some differences highlighted.      

On the Cowen-Douglas Class for Banach Space Operators

In [3], M. J. Cowen and R. G. Douglas prove that the adjoint of a Hilbert space operator T is in the class if and only if T is unitarily equivalent with the operator M _z on a Hilbert space -valued analytic functions, where M _z denotes the operator of multiplication by the independent variable. The proof involves holomorphic vector bundles and Grauert’s theorem. In this paper we use a theorem by I. Gohberg and L. Rodman [4] to give a more elementary proof of this fact, which also works for Banach space operators.      

Non-Commutative Harmonic and Subharmonic Polynomials

The paper introduces a notion of the Laplace operator of a polynomial p in noncommutative variables x = (x ₁,…, x _g ). The Laplacian Lap[p, h] of p is a polynomial in x and in a noncommuting variable h. When all variables commute we have Lap[p, h] = h ²Δ _x p where Δ _x p is the usual Laplacian. A symmetric polynomial in symmetric variables will be called harmonic if Lap[p, h] = 0 and subharmonic if the polynomial q(x, h) :=  Lap[p, h] takes positive semidefinite matrix values whenever matrices X ₁,…, X _g , H are substituted for the variables x ₁,…, x _g , h. In this paper we classify all homogeneous symmetric harmonic and subharmonic polynomials in two symmetric variables. We find there are not many of them: for example, the span of all such subharmonics of any degree higher than 4 has dimension 2 (if odd degree) and 3 (if even degree). Hopefully, the approach here will suggest ways of defining and analyzing other partial differential equations and inequalities. Dedicated to Israel Gohberg on the occasion of his 80^th birthday. All authors were partially supported by J.W. Helton’s grants from the NSF and the Ford Motor Co. and J. A. Hernandez was supported by a McNair Fellowship.     

Existence of Solutions for Second Order Partial Neutral Functional Differential Equations

We establish existence of mild solutions for a class of abstract second-order partial neutral functional differential equations with unbounded delay in a Banach space. The work of the second author was supported by FONDECYT-CONICYT, Grants 1050314 and 7050034.     

Axiomatizing the Harsanyi solution,the symmetric egalitarian solution and the consistent solution for NTU-games

The validity of the axiomatization of the Harsanyi solution for NTU-games by Hart (1985) is shown to depend on the regularity conditions imposed on games. Following this observation, we propose two related axiomatic characterizations, one of the symmetric egalitarian solution (Kalai and Samet, 1985) and one of the consistent solution (Maschler and Owen, 1992). The three axiomatic results are studied, evaluated and compared in detail.Revised October 2004We thank an anonymous referee and an associate editor for their helpful comments. Geoffroy de Clippel also thanks Professors Sergiu Hart, Jean-François Mertens and Enrico Minelli. Horst Zank thanks the Dutch Science Foundation NWO and the British Council for support under the UK-Netherlands Partnership Programme in Science (PPS 706). The usual disclaimer applies.     

Packing Transitive Triples in a Tournament

We prove that a tournament with n vertices has more than arc-disjoint transitive triples, and more than arc-disjoint transitive quadruples, improving earlier bounds. In particular, 82 percent of the arcs of a tournament can be packed with transitive triples and 45 percent of the arcs of a tournament can be packed with transitive quadruples. Our proof is obtained by examining the fractional version of the problem and utilizing a connection between the integral and fractional versions. Received October 12, 2005     

A Construction of Subsets of the Reals which have a Similarity Decomposition

We construct a subset of the set R of real numbers of cardinality |R| which has a similarity decomposition, and which has an ordertype < that of R. Seymour Ginsburg had posed the question whether there exist sets with another ordertype than that of R which also have a similarity decomposition.      

On a pythagorean-type functional equation with exponent 3 involving finite fields

We solve the functional equationf(x ³ +y ³) =f(x)³ +f(y)³ for maps of a finite field into itself.     

Rate of convergence to a singular steady state of a supercritical parabolic equation

We study global positive solutions of a supercritical parabolic equation which converge to a steady state that is singular at x = 0. We determine the rate of convergence to the singular steady state in where B _ν(0) is a ball in with the center at the origin and radius ν.