1. CJM Online first
|On Whitney-type characterization of approximate differentiability on metric measure spaces|
We study approximately differentiable functions on metric measure spaces admitting a Cheeger differentiable structure. The main result is a Whitney-type characterization of approximately differentiable functions in this setting. As an application, we prove a Stepanov-type theorem and consider approximate differentiability of Sobolev, $BV$ and maximal functions.
Keywords:approximate differentiability, metric space, strong measurable differentiable structure, Whitney theorem
Categories:26B05, 28A15, 28A75, 46E35
2. CJM 2011 (vol 63 pp. 460)
| Monotonically Controlled Mappings|
We study classes of mappings between finite and infinite dimensional Banach spaces that are monotone and mappings which are differences of monotone mappings (DM). We prove a RadÃ³-Reichelderfer estimate for monotone mappings in finite dimensional spaces that remains valid for DM mappings. This provides an alternative proof of the FrÃ©chet differentiability a.e. of DM mappings. We establish a Morrey-type estimate for the distributional derivative of monotone mappings. We prove that a locally DM mapping between finite dimensional spaces is also globally DM. We introduce and study a new class of the so-called UDM mappings between Banach spaces, which generalizes the concept of curves of finite variation.
Keywords: monotone mapping, DM mapping, RadÃ³-Reichelderfer property, UDM mapping, differentiability
3. CJM 2005 (vol 57 pp. 961)
|Cone-Monotone Functions: Differentiability and Continuity |
We provide a porosity-based approach to the differentiability and continuity of real-valued functions on separable Banach spaces, when the function is monotone with respect to an ordering induced by a convex cone $K$ with non-empty interior. We also show that the set of nowhere $K$-monotone functions has a $\sigma$-porous complement in the space of continuous functions endowed with the uniform metric.
Keywords:Cone-monotone functions, Aronszajn null set, directionally porous, sets, GÃ¢teaux differentiability, separable space
4. CJM 2004 (vol 56 pp. 825)
|Differentiability Properties of Optimal Value Functions |
Differentiability properties of optimal value functions associated with perturbed optimization problems require strong assumptions. We consider such a set of assumptions which does not use compactness hypothesis but which involves a kind of coherence property. Moreover, a strict differentiability property is obtained by using techniques of Ekeland and Lebourg and a result of Preiss. Such a strengthening is required in order to obtain genericity results.
Keywords:differentiability, generic, marginal, performance function, subdifferential
Categories:26B05, 65K10, 54C60, 90C26, 90C48