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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 Categories:26B05, 46G05 |
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 Categories:26B05, 58C20 |
4. CJM 2004 (vol 56 pp. 699)
Bump Functions with HÃ¶lder Derivatives We study the range of the gradients
of a $C^{1,\al}$-smooth bump function defined on a Banach space.
We find that this set must satisfy two geometrical conditions:
It can not be too flat and it satisfies a strong compactness condition
with respect to an appropriate distance.
These notions are defined precisely below.
With these results we illustrate the differences with
the case of $C^1$-smooth bump functions.
Finally, we give a sufficient condition on a subset of $X^{\ast}$ so that it is
the set of the gradients of a $C^{1,1}$-smooth bump function.
In particular, if $X$ is an infinite dimensional Banach space
with a $C^{1,1}$-smooth bump function,
then any convex open bounded subset of $X^{\ast}$ containing $0$ is the set
of the gradients of a $C^{1,1}$-smooth bump function.
Keywords:Banach space, bump function, range of the derivative Categories:46T20, 26E15, 26B05 |
5. 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 |