BV ‐extension of rate independent operators

Rate independent operators naturally arise in the mathematical analysis of hysteresis. Among rate independent operators, the locally monotone ones are those better suited for the study of PDE's with hysteresis. We prove that a rate independent operator R: Lip (0, T) → BV (0, T) ∩ C (0, T) which is locally monotone and continuous with respect to the strict topology of BV admits a unique continuous extension R?: BV (0, T) → BV (0, T). This general result applies to several concrete hysteresis operators. For many of these operators the existence of a continuous extension was previously known at most to the space BV (0, T) ∩ C (0, T). (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Probabilistic Choice in Games: Properties of Rosenthal’s t-Solutions

The t-solutions introduced in R. W. Rosenthal (1989, Int J Game Theory 18:273–292) are quantal response equilibria based on the linear probability model. Choice probabilities in t-solutions are related to the determination of leveling taxes in taxation problems. The set of t-solutions coincides with the set of Nash equilibria of a game with quadratic control costs. Evaluating the set of t-solutions for increasing values of t yields that players become increasingly capable of iteratively eliminating never-best replies and eventually only play rationalizable actions with positive probability. These features are not shared by logit quantal response equilibria. Moreover, there exists a path of t-solutions linking uniform randomization to Nash equilibrium     

The conjugate heat equation and Ancient solutions of the Ricci flow

We prove Gaussian type bounds for the fundamental solution of the conjugate heat equation evolving under the Ricci flow. As a consequence, for dimension 4 and higher, we show that the backward limit of Type I κ-solutions of the Ricci flow must be a non-flat gradient shrinking Ricci soliton. This extends Perelman?s previous result on backward limits of κ-solutions in dimension 3, in which case the curvature operator is nonnegative (it follows from Hamilton–Ivey curvature pinching estimate). As an application, this also addresses an issue left in Naber (2010) [23], where Naber proves the interesting result that there exists a Type I dilation limit that converges to a gradient shrinking Ricci soliton, but that soliton might be flat. The Gaussian bounds that we obtain on the fundamental solution of the conjugate heat equation under evolving metric might be of independent interest.     

On Backward Uniqueness for Parabolic Equations

We prove a backward uniqueness result for the heat operator with variable lower order terms, which implies the full regularity of L _3,-solutions of the three-dimensional Navier–Stokes equations. Bibliography: 12 titles.     

Self-adjoint elliptic operators and manifold decompositions part I: Low Eigenmodes and stretching

This paper is the first of a three-part investigation into the behavior of analytical invariants of manifolds that can be split into the union of two submanifolds. In this article, we will show how the low eigensolutions of a self-adjoint elliptic operator over such a manifold can be studied by a splicing construction. This construction yields an approximated solution of the operator whenever we have two L²-solutions on both sides and a common limiting value of two extended L²-solutions. In Part II, the present analytic “Mayer-Vietoris” results on low eigensolutions and further analytic work will be used to obtain a decomposition theorem for spectral flows in terms of Maslov indices of Lagrangians. In Part III after comparing infinite- and finite-dimensional Lagrangians and determinant line bundles and then introducing “canonical perturbations” of Lagrangian subvarieties of symplectic varieties, we will study invariants of 3-manifolds, including Casson's invariant. © 1996 John Wiley & Sons, Inc.     

Positive Operators à la Aumann-Shapley on Spaces of Functions on D-Lattices

Let L be a σ-complete D-lattice and BV the AL-space of all realvalued, null in zero, functions on L of bounded variation. We prove the existence of a continuous Aumann-Shapley operator on the closed subspace of BV generated by powers of nonatomic σ-additive positive modular measures on L. The integral representation of on a class of functions that correspond to measure games is also exhibited. Dedicated to Professor Paolo De Lucia on the occasion of his 70th birthday.     

Uniformly continuous superposition operators in the space of bounded variation functions

Let I, J ? ? be intervals. The main result says that if a superposition operator H generated by a function of two variables h: I × J → ?, H (φ)(x) ? h (x, φ (x)), maps the set BV (I, J) of all bounded variation functions, φ: I → J into the Banach space BV (I, ?) and is uniformly continuous with respect to the BV ‐norm, then h (x, y) = a (x)y + b (x), x ∈ I, y ∈ J, for some a, b ∈ BV (I, ?) (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Generalized solutions for a class of non-uniformaly elliptic equations in divergence form

We study a general class of quasilinear non-uniformly elliptic pdes in divergence from with linear growth in the gradient. We examine the notions of BV and viscosity solutions and derive for such generalized solutions various a priori pointwise and integral estimates, including a Harnack inequality. In particular we prove that viscosity solutions are unique (on strictly convex domains), are contained in the space BV ^loc and are C ^1,α almost everywhere.     

Removable sets for homogeneous linear partial differential equations in Carnot groups

Let L be a homogeneous left-invariant differential operator on a Carnot group. Assume that both L and L^t are hypoelliptic. We study the removable sets for L-solutions. We give precise conditions in terms of the Carnot- Caratheodory Hausdorff dimension for the removability for L-solutions under several auxiliary integrability or regularity hypotheses. In some cases, our criteria are sharp on the level of the relevant Hausdorff measure. One of the main ingredients in our proof is the use of novel local self-similar tilings in Carnot groups.     

Lower semicontinuity of quasi-convex integrals in BV

Some boundedness properties for an extension operator are proved and used together with techniques of Maly [24], Meyers [29], Fonseca and Müller [13] and Fonseca and Marcellini [12] to obtain lower semicontinuity results in BV for quasiconvex integrals of super-linear growth. Received January 25, 1997 / Accepted October 3, 1997     

Periodic solutions of the stefan problem with hysteresis-type boundary conditions

We consider the Stefan problem with Dirichlet boundary conditions depending on a hysteresis functional where the free boundary is involved. We show existence of a positive valueT and existence of aT-periodic solution of the problem, provided the Stefan number is sufficiently small and the hysteresis functional is described by the elementary rectangular hysteresis loop. If in addition the Preisach hysteresis operator is Lipschitz-continuous we prove that every periodic solution must be stationary. Dedicated to Professor Avner Friedman on occasion of his 60th birthday supported by Humboldt Foundation Scholarship     

Integral self-affine sets with positive Lebesgue measures

A self-affine region is an integral self-affine set with positive Lebesgue measure. In this note we give two criteria for integral self-affine sets being self-affine regions. As their applications we study the L ¹-solutions of refinement equations, which play an important role in constructing wavelets, and we give several interesting examples. Received: 8 May 2007     

Hysteresis nonstationary nonlinearities

We consider an operator (variable hysteron) used to describe a nonstationary hysteresis nonlinearity (whose characteristics vary under the action of external forces) according to the Krasnosel’skii-Pokrovskii scheme. Sufficient conditions under which the operator is defined for the inputs from the class of functions H ¹[t ₀, T] satisfying the Lipschitz condition in the segment [t ₀, T] are established. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 3, pp. 295–309, March, 2008.     

Sobolev and strict continuity of general hysteresis operators

The most natural and important topologies connected with hysteresis operators are those induced by uniform convergence, W^{1, 1}‐convergence, and strict convergence. Indeed the supremum norm and the variation are invariant under reparameterization. We prove a general result that implies that if a hysteresis operator is continuous with respect to the topology of W^{1, 1}, then it is continuous with respect to the strict topology. Copyright © 2009 John Wiley & Sons, Ltd.     

A two-dimensional Gauss-Kuzmin theorem for singular continued fractions

A very simple proof of a generalization of the Gauss-Kuzmin theorem for singular continued fractions is given by considering the transition operator defined in [Se] as an operator on the Banach space BV(W) of complex-valued functions of bounded variation on W = [0, ( )]. The upper bound obtained here implies that the convergence rate, O(αⁿ), with 0.17 ≤ α ≤ 0.47 < g, is better than that obtained in [DK].     

On the Uniform Convergence of Walsh-Fourier Series

We study the uniform convergence of Walsh-Fourier series of functions on the generalized Wiener class BV (p(n)↑∞) This revised version was published online in June 2006 with corrections to the Cover Date.     

Redistribution of Damping in Viscoelasticity

Physically admissible, global BV solutions are constructed to the Cauchy problem for the equations of one-dimensional, nonlinear viscoelasticity of the Boltzmann type. The required BV bounds are derived with the help of energy estimates, in conjunction with a change of variables that redistributes the damping on an equitable basis among the equations of the system, thus exposing the dissipative effect of viscosity on the variation of solutions.