Local Lipschitz continuity of the stop operator

On a closed convex set Z in ^N with sufficiently smooth (W ^2,) boundary, the stop operator is locally Lipschitz continuous from W ^1,1([0,T]^N) × Z into W ^1,1([0,T],^N). The smoothness of the boundary is essential: A counterexample shows that C ¹-smoothness is not sufficient.     

Hyperbolic systems of conservation laws with Lipschitz continuous flux-functions: The Riemann problem

For strictly hyperbolic systems of conservation laws with Lipschitz continuous flux-functions we generalize Lax's genuine nonlinearity condition and shock admissibility inequalities and we solve the Riemann problem when the left- and right-hand initial data are sufficiently close. Our approach is based on the concept of multivalued representatives ofL functions and a generalized calculus for Lipschitz continuous mappings. Several interesting features arising with Lipschitz continuous flux-functions come to light from our analysis.Dedicated to Constantine Dafermos on his 60^th birthday     

A Note on Euler's Approximations

We prove that Euler's approximations for stochastic differential equations on domains of ^d converge almost surely if the drift satisfies the monotonicity condition and the diffusion coefficient is Lipschitz continuous.     

Admissibility and Observability of Observation Operators for Semilinear Problems

This paper deals with semilinear evolution equations with unbounded observation operators. Sufficient conditions are given guaranteeing that the output function of a semilinear system is in L²_loc([0, ∞); Y). We prove that the Lebesgue extension of the observation operators are invariant under nonlinear globally Lipschitz continuous perturbations. Further, relations between the corresponding -extensions are studied. We show that exact observability of linear autonomous system is conserved under small Lipschitz perturbations. The obtained results are illustrated by several examples.      

A product integration method for a class of singular first kind Volterra equations

Summary Convergence of a midpoint product integration method for singular first kind Volterra equations with kernels of the formk(t, s)(t–s) ^– , 0<<1, wherek(t, s) is continuous, is examined. It is shown that convergence of order one holds if the solution of the Volterra equation has a Lipschitz continuous first derivative andk(t, s) is suitably smooth. In addition, convergence is shown to hold when the solution has only Lipschitz continuity and the same conditions onk(t, s) apply. An existence theorem of Kowalewski is used to relate these conditions on the solution to conditions on the data andk(t, s).     

Lipschitz continuity of the best approximation operator in vector-valued Chebyshev approximation

When G is a finite dimensional Haar subspace of C(X,R^k), the vector-valued continuous functions (including complex-valued functions when k is 2) from a finite set X to Euclidean k-dimensional space, it is well-known that at any function f in C(X,R^k) the best approximation operator satisfies the strong unicity condition of order 2 and a Lipschitz (Hőlder) condition of order . This note shows that in fact the best approximation operator satisfies the usual Lipschitz condition of order 1.     

Implicit function theorems for generalized equations

We show that Lipschitz and differentiability properties of a solution to a parameterized generalized equation 0 f(x, y) + F(x), wheref is a function andF is a set-valued map acting in Banach spaces, are determined by the corresponding Lipschitz and differentiability properties of a solution toz g(x) + F(x), whereg strongly approximatesf in the sense of Robinson. In particular, the inverse map (f + F)^–1 has a local selection which is Lipschitz continuous nearx ₀ and Fréchet (Gateaux, Bouligand, directionally) differentiable atx ₀ if and only if the linearization inverse (f (x ₀) + f (x₀) (× – x₀) + F(×))^–1 has the same properties. As an application, we study directional differentiability of a solution to a variational inequality.This work was supported by National Science Foundation Grant Number DMS 9404431.     

Quantitative stability of variational systems: III.ε-approximate solutions

We prove that the-optimal solutions of convex optimization problems are Lipschitz continuous with respect to data perturbations when these are measured in terms of the epi-distance. A similar property is obtained for the distance between the level sets of extended real valued functions. We also show that these properties imply that the-subgradient mapping is Lipschitz continuous.Research supported in part by the National Science Foundation and the Air Force Office of Scientific Research.     

On the martingale problem for Banach space valued stochastic differential equations

LetE denote a real separable Banach space and letZ=(Z(t, f) be a family of centered, homogeneous, Gaussian independent increment processes with values inE, indexed by timet0 and the continuous functionsf:[0,t] E. If the dependence ont andf fulfills some additional properties,Z is called a gaussian random field. For continuous, adaptedE-valued processesX a stochastic integral processY = ₀ ^. Z(t, X)(dt) is defined, which is a continuous local martingale with tensor quadratic variation[Y] = ₀ ^. Q(t, X)dt, whereQ(t, f) denotes the covariance operator ofZ(t, f).Y is called a solution of the homogeneous Gaussian martingale problem, ifY = ₀ ^. Z(t, Y)(dt). Such solutions occur naturally in connection with stochastic differential equations of the type (D):dX(t)=G(t, X) dt+Z(t, X)(dt), whereG is anE-valued vector field. It is shown that a solution of (D) can be obtained by a kind of variation of parameter method, first solving a deterministic integral equation only involvingG and then solving an associated homogeneous martingale problem.     

On (p a , p b , p a , p a-b )-Relative Difference Sets

This paper provides new exponent and rank conditions for the existence of abelian relative (p ^a,p ^b,p ^a,p ^a–b)-difference sets. It is also shown that no splitting relative (2^2c,2^d,2^2c,2^2c–d)-difference set exists if d > c and the forbidden subgroup is abelian. Furthermore, abelian relative (16, 4, 16, 4)-difference sets are studied in detail; in particular, it is shown that a relative (16, 4, 16, 4)-difference set in an abelian group G Z₈ × Z₄ × Z₂ exists if and only if exp(G) 4 or G = Z₈ × (Z₂)³ with N Z₂ × Z₂.     

Implicit functions and sensitivity of stationary points

We consider the spaceL(D) consisting of Lipschitz continuous mappings fromD to the Euclideann-space ⁿ ,D being an open bounded subset of ⁿ . LetF belong toL(D) and suppose that solves the equationF(x) = 0. In case that the generalized Jacobian ofF at is nonsingular (in the sense of Clarke, 1983), we show that forG nearF (with respect to a natural norm) the systemG(x) = 0 has a unique solution, sayx(G), in a neighborhood of Moreover, the mapping which sendsG tox(G) is shown to be Lipschitz continuous. The latter result is connected with the sensitivity of strongly stable stationary points in the sense of Kojima (1980); here, the linear independence constraint qualification is assumed to be satisfied.     

Global Lipschitz continuity of free boundaries in the one-phase Stefan problem

Summary It is proved that the free boundary t=s(x)in the one-phase Stefan problem is uniformly Lipschitz continuous in the strip 0tT for any T>0with a Lipschitz coefficient depending only on the specified data under some conditions.This work was supported by C.N.R. Italy.     

On directional entropy functions

Given aZ ²-process, the measure theoretic directional entropy function,h( % MathType!End!2!1!), is defined on % MathType!End!2!1!. We relate the directional entropy of aZ ²-process to itsR ² suspension. We find a sufficient condition for the continuity of directional entropy function. In particular, this shows that the directional entropy is continuous for aZ ²-action generated by a cellular automaton; this finally answers a question of Milnor [Mil]. We show that the unit vectors whose directional entropy is zero form aG _δ subset ofS ¹. We study examples to investigate some properties of directional entropy functions. This research is supported in part by BSRI and KOSEF 95-0701-03-3.     

Smallest g -supersolution with constraint

In this paper the existence and uniqueness of the smallest g-supersolution for BSDE is discussed in the case without Lipschitz condition imposing on both constraint function and drift coefficient in the different method from the one with Lipschitz condition. Then by considering (ξ, g) as a parameter of BSDE, and (ξ ^α, g ^α) as a class of parameters for BSDE, where α belongs to a set , for every there exists a pair of solution {Y ^a, Z^a} for the BSDE, the properties of which is also a solution for some BSDE is studied. This result may be used to discuss optimal problems with recursive utility. This work was supported by NSFC (79790130)     

Energy of Flows on Percolation Clusters

It is well known for which gauge functions H there exists a flow in Z ^d with finite H energy. In this paper we discuss the robustness under random thinning of edges of the existence of such flows. Instead of Z ^d we let our (random) graph cal C cal (Z ^d,p) be the graph obtained from Z ^d by removing edges with probability 1–p independently on all edges. Grimmett, Kesten, and Zhang (1993) showed that for d3,p>p _c(Z ^d), simple random walk on cal C cal (Z ^d,p) is a.s. transient. Their result is equivalent to the existence of a nonzero flow f on the infinite cluster such that the x ² energy _e f(e)² is finite. Levin and Peres (1998) sharpened this result, and showed that if d3 and p>p _c(Z ^d), then cal C cal (Z ^d,p) supports a nonzero flow f such that the x ^q energy is finite for all q>d/(d–1). However, for general gauge functions, there is a gap between the existence of flows with finite energy which results from the work of Levin and Peres and the known results on flows for Z ^d. In this paper we close the gap by showing that if d3 and Z ^d supports a flow of finite H energy then the infinite percolation cluster on Z ^d also support flows of finite H energy. This disproves a conjecture of Levin and Peres.     

Subelliptic Estimates for a Class of Degenerate Elliptic Integro-Differential Operators

In this paper we prove subelliptic estimates for operators of the form Δ_x + λ² (x)S in ?^N = ? × ?, where the operator S is an elliptic integro - differential operator in ?^N and λ is a nonnegative Lipschitz continuous function.     

The Automorphism Groups of Steiner Triple Systems Obtained by the Bose Construction

The automorphism group of the Steiner triple system of order v 3 (mod 6), obtained from the Bose construction using any Abelian Group G of order 2s + 1, is determined. The main result is that if G is not isomorphic to Z ₃ ⁿ × Z ₉ ^m , n 0, m 0, the full automorphism group is isomorphic to Hol(G) × Z ₃, where Hol(G) is the Holomorph of G. If G is isomorphic to Z ₃ ⁿ × Z ₉ ^m , further automorphisms occur, and these are described in full.     

Interrelations between the Length, the Structure and the Cardinality of a Chain

Let (Z,) be a chain and let (Z,) be its dual chain. Then the length l(Z) of (Z,) is the least upper bound of all cardinal numbers which can be order-embedded into (Z,) or (Z,). In particular, a chain is said to be short if its length is not greater than the smallest infinite cardinal. In this paper we shall prove that the cardinality |Z| of a chain (Z,) cannot be smaller than l(Z) and not greater than 2 ^l(Z). The inequality |Z|2 ^l(Z) is an immediate consequence of a general theorem which combines the structure of a chain with its length. In case of a short chain it follows that its structure may be rather complicated but that its cardinality cannot be greater than the cardinality of the real line.     

Binding energy for hydrogen-like atoms in the Nelson model without cutoffs

In the Nelson model particles interact through a scalar massless field. For hydrogen-like atoms there is a nucleus of infinite mass and charge Ze, Z>0, fixed at the origin and an electron of mass m and charge e. This system forms a bound state with binding energy E_bin=me⁴Z²/8π² to leading order in e. We investigate the radiative corrections to the binding energy and prove upper and lower bounds which imply that with explicit coefficient c₀ and independent of the ultraviolet cutoff. c₀ can be computed by perturbation theory, which however is only formal since for the Nelson Hamiltonian the smallest eigenvalue sits exactly at the bottom of the continuous spectrum.     

Uniform Hölder Bounds for Nonlinear Schrödinger Systems with Strong Competition

For the positive solutions of the Gross–Pitaevskii system we prove that L^∞‐boundedness implies C^0,α‐boundedness for every α ? (0,1), uniformly as β → +∞. Moreover, we prove that the limiting profile as β → +∞ is Lipschitz‐continuous. The proof relies upon the blowup technique and the monotonicity formulae by Almgren and Alt, Caffarelli, and Friedman. This system arises in the Hartree‐Fock approximation theory for binary mixtures of Bose–Einstein condensates in different hyperfine states. Extensions to systems with k > 2 densities are given. © 2009 Wiley Periodicals, Inc.