Pathological Lipschitz Functions In $R^{N}$

Wang, Shawn Xianfu (1995) Pathological Lipschitz Functions In $R^{N}$. [Preprint]

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Abstract

In recent years four subdifferential maps have been widely used: the {\bf Clarke subdifferential}, the {\bf Michel--Penot subdifferential}, the {\bf Ioffe--Mordu-khovich--Kruger approximate subdifferential}, and the {\bf Dini subdifferential}. We denote these four notions by C', MP', A', and D' respectively. Each of them is a generalization from convex to locally Lipschitz functions and each of them generalizes different aspects of the convex situation. In this thesis, we construct Lipschitz functions with pathological properties and study the differences among these four subdifferential maps. \par Chapter 1 contains some basic concepts and notation from nonsmooth analysis. These are: subdifferentiability, subderivative, minimal usco, minimal cusco, regularity, integrability, and saineness of a Lipschitz function.\par In Chapter 2, we consider our subdifferential maps on the real line. For differentiable functions we prove that most functions have C--subdifferentials which are singleton almost nowhere and most have C--subdifferentials and MP--subdifferentials which differ almost everywhere. We also show that C--integrability and A--integrability coincide on the real line for any locally Lipschitz function. We then show that the C--subdifferential and A--subdifferential can be different at most on a countable set for any locally Lipschitz function. Finally using Borwein and Fitzpatrick's theorem we construct Lipschitz functions which are regular on sets with small measure.\par In Chapter 3, we consider the inverse problem. We provide a technique for constructing Lipschitz functions with prescribed subdifferentials. More precisely we show that if $f_{1}$, $f_{2}$,..., $f_{k}$ possess minimal C--subdifferential mappings on an open set $U$, then there exists a real--valued locally Lipschitz function $g$ defined on $U$ such that: $$\partial_{c}g(x)=conv\{\partial_{c}f_{1}(x),\partial_{c}f_{2}(x), \cdots, \partial_{c}f_{k}(x)\}\mbox{\quad for each x\in U}.$$ As a result of this, we deduce that given a finite family of maximal {\bf cyclically monotone} operators $\{T_{1}, T_{2},..., T_{k}\}$ on an open set $U$ there exists a real--valued locally Lipschitz function $g$ defined on $U$ such that: $$\partial_{c}g(x)=conv\{T_{1}(x),T_{2}(x), \cdots, T_{k}(x)\}\mbox{ \quad for each x\in U}.$$ In particular, we obtain that given any convex polytope $P$ in a finite dimensional space $X$ there is a locally Lipschitz function $f$ such that $\partial_{c}f(x)=P$ for each $x\in U$. This shows that without restrictions, the C--subdifferential of a locally Lipschitz function can be a somewhat unwieldy beast. \par In Chapter 4, we investigate {\bf bump functions}. We begin by showing that any strictly convex body containing 0 is the gradient range of a smooth bump function and that the ranges of C--subdifferentials and A--subdifferentials always contain 0 as an interior point. We use bump functions to construct a Lipschitz function in $R^{2}$ such that its C--subdifferential and A--subdifferential differ on a set with large measure. Then we show that there is a Lipschitz function in $R^{2}$ such that its A--subdifferential has nonconvex images almost everywhere. Finally we construct two Lipschitz functions with the same C--subdifferential but with their A--subdifferentials differing except on a set with small measure.

Item Type: Preprint pubdom FALSE subdifferential, differentiability, minimal usco, minimal cusco, maximal monotone operator, maximal cyclically monotone operator, regularity, integrability, sameness, metric density, approximate continuity, Hausdorff metric, gauge, polar, bump function 58-xx Global analysis, analysis on manifolds > 58Cxx Calculus on manifolds; nonlinear operators52-xx Convex and discrete geometry > 52Axx General convexity26-xx Real functions > 26Axx Functions of one variable49-xx Calculus of variations and optimal control; optimization > 49Jxx Existence theories46-xx Functional analysis > 46Nxx Miscellaneous applications of functional analysis UNSPECIFIED Users 1 not found. 17 Nov 2003 21 Apr 2010 11:13 https://docserver.carma.newcastle.edu.au/id/eprint/113

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