In his paper, the concepts of metric curvature and folding of a Cl-representable manifold in a normed linear space are studied. With certain restrictions on the metric curvature and/or folding, one can obtain a neighborhood of unique best approximation from the manifold, and in some cases, the manifold can be shown to be Chebyshev. Several familiar examples, including some classes of T-polynomials, are given.1. Introduction. The purpose of this paper is to study unique best approximation from subsets of normed linear spaces which are Cl-manifolds with boundary. In order to study such problems from a general geometric vantage point, John R. Rice [12] introduced the concepts of folding and metric curvature (originally called curvature) in smooth, rotund, and finite-dimensional spaces. These concepts were generalized to uniformly smooth spaces by two of the authors in [13] (see also [5], [6]). In 3 and 4 of the present work, we closely examine these concepts for Cl-manifolds with boundaries and obtain results which demonstrate several connections between local uniqueness, metric curvature, and folding.In C[a, b], there are nonlinear sets, for instance, the set of rational functions of degree no greater than (m, n), that are Chebyshev. We show that the fact that Haar embedded manifolds are Chebyshev (proved in a special case by Daniel Wulbert 17] and generalized by D. Braess [2]) follows from general results on the metric curvature of the manifold (see Theorem 6.1). For example, we see in 7 that the set N is Chebyshev in C[a, b], where 0 < a < b.In L([a, b], x) it is well known [14, p. 368] that a nonconvex boundedly compact subset is not Chebyshev (i.e., there is a point which does not have a unique best approximation from the set). In 7 we exhibit for the first time some familiar subsets M of L2 ([a, b],/x) each of which is a Cl-manifold with boundary and has a neighborhood of unique best approximation for M. Thus, for points in this neighborhood of M, steepest descent methods may be attempted. Many nonlinear regression problems fall into the above category.Both of the above results are special cases of Theorem 5.1 in this paper, which basically states that every manifold M with boundary which has finite metric curvature and positive folding has a neighborhood of unique best approximation from M. * Received by the editors August 16, 1974.
Let B be a fiber bundle with compact fiber F over a compact Riemannian n-manifold M n . There is a natural Riemannian metric on the total space B consistent with the metric on M. With respect to that metric, the volume of a rectifiable section σ : M → B is the mass of the image σ (M) as a rectifiable n-current in B. (2000): 49F20, 49F22, 49F10, 58A25, 53C42, 53C65. Theorem 1. For any homology class of sections of B, there is a mass-minimizing rectifiable current T representing that homology class which is the graph of a C 1 section on an open dense subset of M. Mathematics Subject Classifications
Abstract. Let M be a compact, oriented Riemannian nmanifold, and let B → M be a fiber bundle over M , with compact fiber F . Given a section of B, its volume is defined as the n-dimensional Hausdorff measure (or mass) of the graph of ϕ. We show that a volume-minimizing graph in a given homology class, which is in general a rectifiable section, is a continuous graph over all but a set of Hausdorff codimension 3 in the base M , unless the fiber F admits stable 2-dimensional minimal currents (without boundary), in which case there could be a codimension 2 set of singular points with poles (support of the current over a singular point) consisting of such 2-dimensional currents. Locally, we also show that a volume-minimizing graph of a map ϕ : Ω → F for Ω ⊂ R n with prescribed boundary ϕ |∂Ω = ψ : ∂Ω → F has codimension-3 poles in Ω, under the assumption that Ω is convex and that F is a compact, convex subset of Euclidean space.This work grew out of a question posed by Herman Gluck and Wolfgang Ziller regarding the "volume of flows", volumes of sections of the unit tangent bundle of a compact manifold, viewed as one-dimensional foliations on M . They established that the standard Hopf fibration, as a foliation on S 3 , minimizes volume among all such sections. However, in higher dimensions the Hopf fibrations are not volume-minimizing, and it is likely that volume-minimizing flows on these manifolds are singular, due to examples of Sharon Pedersen.In this setting our results show that, except in dimension 3, outside of a codimension-3 set of possible "poles", a minimizing foliation will be C 1 . In dimension 3 there can be a codimension-2 set of poles, almost all of which involve the whole fiber in that for each unit tangent vector at almost-all poles (in the Hausdorff measure on the set of singular points), there is a limit of tangents of a leaf converging to that vector.
In part I (P. Smith, Perron's method for quasilinear hyperbolic systems, part I, J. Math. Anal., in press) of this paper we defined a notion of viscosity solution (sub-(super-)solution) for these systems, proved a comparison principle for viscosity sub-and supersolutions. Here, in part II, we prove existence of viscosity solutions to the Cauchy problem, using a Perron-like method, for long time, and for all time. 2005 Elsevier Inc. All rights reserved.
Convergence of Rothe's method for the fully nonlinear parabolic equation u t + F (D 2 u, Du, u, x, t) = 0 is considered under some continuity assumptions on F. We show that the Rothe solutions are Lipschitz in time, and they solve the equation in the viscosity sense. As an immediate corollary we get Lipschitz behavior in time of the viscosity solutions of our equation.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.