Linear second-order divergence equations in Lipschitz domains

In this paper we generalize gradient estimates in Lp spaces to Orlicz spaces for weak solutions of second-order divergence elliptic equations with small BMO coefficients in Lipschitz domains. Our results improve the known results for such equations using the harmonic analysis method.     

An effective model for Lipschitz wrinkled arches

Within the framework of the Koiter's linear elastic shell theory, we study the limit model of a Lipschitz curved arch whose mid-surface is periodically waved. The magnitude and the period of the wavings are of the same order. To achieve the asymptotic analysis, we consider a mixed formulation, for which we perform a two-scale homogenization technique. We prove the convergence of the displacements, the rotation of the normal, and the membrane strain. From the limit formulation, we derive an effective model for curved critically wrinkled arches. It introduces two membrane strain functions—instead of one in the classical case—and exhibits a corrector membrane term to the coupling between the rotation of the normal and the membrane strain.     

Linear O-minimal structures

A linearly ordered structure is called o-minimal if every definable subset ofM is a finite union of points and intervals. Such an is aCF structure if, roughly said, every definable family of curves is locally a one-parameter family. We prove that if is aCF structure which expands an (interval in an) ordered group, then it is elementary equivalent to a reduct of an (interval in an) ordered vector space. Along the way we prove several quantifier-elimination results for expansions and reducts of ordered vector spaces. The research for this article was begun when the authors were at Berkeley during the logic year at the Mathematical Science Research Institute. It was completed at McGill University. The research was supported by grants from NSERC and FCAR. JL would, as always, like to thank Alistair.     

群零模正则化问题的等价Lipschitz优化模型

针对群零模正则化问题, 从零模函数的变分刻画入手, 将其等价地表示为带有 互补约束的数学规划问题(简称MPCC问题), 然后证明将互补约束直接罚到MPCC的目标函数而得到的罚问题是MPCC问题的全局精确罚. 此精确罚问题的目标函数不仅在可行集上全局Lipschitz连续而且还具有满意的双线性结构, 为设计群零模正则化问题的序列凸松弛算法提供了满意的等价Lipschitz优化模型.     

Lipschitz stability criteria for a generalized delayed Kolmogorov model

Lipschitz stability and Lipschitz? _° - equistability of the functional differential equation $x' = B(x)f(t,x,x_t ), x_{t_ \circ = \theta _ \circ } $ are discussed. Sufficient conditions are given using the comparison with the corresponding scalar equation.     

Quasi‐Lipschitz condition in potential theory

The velocity of an incompressible flow in a bounded three‐dimensional domain is represented by its vorticity with the help of an apparently new representation formula. Using this formula we prove a quasi‐Lipschitz estimate for in dependence of the supremum norm of . Our quasi‐Lipschitz bound extends to the case where is represented by any continuous ≠ rot     

Lipschitz functions of self-adjoint operators in perturbation theory

Let A be a self-adjoint operator in a Hubert space. In order that for each differentiable function f and for each self-adjoint operator B one should have the estimate f(B)–f(A) c_f B–A it is necessary and sufficient that the spectrum. of the operator A be a finite set. If m is the number of points of the spectrum of the operator A, then for the constant cf one can take 8(log₂m+2)² [f], where [f] is the Lipschitz constant of the function f.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 141, pp. 176–182, 1985.     

Transmission problems and spectral theory for singular integral operators on Lipschitz domains

We prove the well-posedness of the transmission problem for the Laplacian across a Lipschitz interface, with optimal non-tangential maximal function estimates, for data in Lebesgue and Hardy spaces on the boundary. As a corollary, we show that the spectral radius of the (adjoint) harmonic double layer potential K∗ in is less than , whenever is a bounded convex domain and 1<p?2.     

Cotorsion pairs, model category structures, and representation theory

We make a general study of Quillen model structures on abelian categories. We show that they are closely related to cotorsion pairs, which were introduced by Salce [Sal79] and have been much studied recently by Enochs and coauthors [EJ00]. This gives a method of constructing model structures on abelian categories, which we illustrate by building two model structures on the category of modules over a (possibly noncommutative) Gorenstein ring. The homotopy category of these model structures is a generalization of the stable module category much used in modular representation theory. This stable module category has also been studied by Benson [Ben97]. Received: 14 December 2000; in final form: 17 December 2001 / Published online: 5 September 2002     

Mathematical problems of the theory of elasticity of chiral materials for Lipschitz domains

By the potential method, we investigate the Dirichlet and Neumann boundary value problems of the elasticity theory of hemitropic (chiral) materials in the case of Lipschitz domains. We study properties of the single‐ and double‐layer potentials and of certain, generated by them, boundary integral operators. These results are applied to reduce the boundary value problems to the equivalent first and the second kind integral equations and the uniqueness and existence theorems are proved in various function spaces. Copyright © 2005 John Wiley & Sons, Ltd.     

Linear dynamic approximation theory

We study in detail the behavior of some known learning algorithms. We estimate the sum of the squares of the absolute relative errors of some general linear learning algorithms and the sum of the squares of the coefficients obtained by the perceptron algorithm. We prove the convergence of a statistical learning algorithm. The possibility of applications of this theory to biology is discussed.     

On the Lipschitz Modulus of the Argmin Mapping in Linear Semi-Infinite Optimization

This paper is devoted to quantify the Lipschitzian behavior of the optimal solutions set in linear optimization under perturbations of the objective function and the right hand side of the constraints (inequalities). In our model, the set indexing the constraints is assumed to be a compact metric space and all coefficients depend continuously on the index. The paper provides a lower bound on the Lipschitz modulus of the optimal set mapping (also called argmin mapping), which, under our assumptions, is single-valued and Lipschitz continuous near the nominal parameter. This lower bound turns out to be the exact modulus in ordinary linear programming, as well as in the semi-infinite case under some additional hypothesis which always holds for dimensions n ? 3. The expression for the lower bound (or exact modulus) only depends on the nominal problem’s coefficients, providing an operative formula from the practical side, specially in the particular framework of ordinary linear programming, where it constitutes the sharp Lipschitz constant. In the semi-infinite case, the problem of whether or not the lower bound equals the exact modulus for n > 3 under weaker hypotheses (or none) remains as an open problem.     

Regularity and index theory for Dirac–Schrödinger systems with Lipschitz coefficients

Dirac–Schrödinger systems play a central role when modeling Dirac bundles and Dirac–Schrödinger operators near the boundary, along ends or near other singularities of Riemannian manifolds. In this article we develop the Fredholm theory of Dirac–Schrödinger systems with Lipschitz coefficients.     

The Dirichlet and Regularity Problems for Some Second Order Linear Elliptic Systems on Bounded Lipschitz Domains

In this paper, we investigate divergence-form linear elliptic systems on bounded Lipschitz domains in \(\mathbb {R}^{d+1}, d \ge 2\), with L² boundary data. The coefficients are assumed to be real, bounded, and measurable. We show that when the coefficients are small, in Carleson norm, compared to one that is continuous on the boundary, we obtain solvability for both the Dirichlet and regularity boundary value problems given that the coefficients satisfy a certain “pseudo-symmetry” condition.