1. CMB Online first
2. CMB Online first
 Steinerberger, Stefan

An Endpoint Alexandrov Bakelman Pucci estimate in the Plane
The classical AlexandrovBakelmanPucci estimate
for the Laplacian states
$$ \max_{x \in \Omega}{ u(x)} \leq \max_{x \in \partial \Omega}{u(x)}
+ c_{s,n} \operatorname{diam}(\Omega)^{2\frac{n}{s}}
\left\ \Delta u
\right\_{L^s(\Omega)}$$
where $\Omega \subset \mathbb{R}^n$, $u \in C^2(\Omega) \cap
C(\overline{\Omega})$ and $s \gt n/2$. The inequality fails for
$s = n/2$. A Sobolev embedding result of Milman and Pustylnik,
originally phrased in a slightly different context, implies an
endpoint inequality: if $n \geq 3$ and $\Omega \subset \mathbb{R}^n$
is bounded, then
$$ \max_{x \in \Omega}{ u(x)} \leq \max_{x \in \partial \Omega}{u(x)}
+ c_n
\left\ \Delta u
\right\_{L^{\frac{n}{2},1}(\Omega)},$$
where $L^{p,q}$ is the Lorentz space refinement of $L^p$. This
inequality fails for $n=2$ and we prove a sharp substitute result:
there exists $c\gt 0$ such that for all $\Omega \subset \mathbb{R}^2$
with finite measure
$$ \max_{x \in \Omega}{ u(x)} \leq \max_{x \in \partial \Omega}{u(x)}
+ c \max_{x \in \Omega} \int_{y \in \Omega}{ \max
\left\{ 1, \log{
\left(\frac{\Omega}{\xy\^2}
\right)}
\right\}
\left \Delta u(y)
\right dy}.$$
This is somewhat dual to the classical TrudingerMoser inequality;
we also note that it is sharper than the usual estimates given
in Orlicz spaces, the proof is rearrangementfree.
The Laplacian can be replaced by any uniformly elliptic operator
in divergence form.
Keywords:AlexandrovBakelmanPucci estimate, second order Sobolev inequality, TrudingerMoser inequality Categories:35A23, 35B50, 28A75, 49Q20 

3. CMB 2017 (vol 61 pp. 495)
 Chang, DerChen; Yang, Nanping; Wu, HsiChun

PoincarÃ© Lemma on QuaternionLike Heisenberg Groups
For smooth functions $a_1, a_2, a_3, a_4$ on a quaternion Heisenberg
group, we characterize
the existence of solutions of the partial differential operator
system $X_1f=a_1, X_2f=a_2, X_3f=a_3,$ and $X_4f=a_4$.
In addition, a formula for the solution function $f$ is deduced
provided the solvability of the system.
Keywords:bracket generating property, quaternion Heisenberg group, curl, integrability condition, PoincarÃ© lemma Categories:93B05, 49N99 

4. CMB 2017 (vol 60 pp. 631)
 ShahrokhiDehkordi, M. S.

Traceless Maps as the Singular Minimizers in the Multidimensional Calculus of Variations
Let $\Omega \subset \mathbb{R}^n$ be a bounded Lipschitz domain and
consider
the energy functional
\begin{equation*}
{\mathcal F}[u, \Omega] := \int_{\Omega} {\rm F}(\nabla {\bf u}(\bf x))\, d{\bf x},
\end{equation*}
over the space of $W^{1,2}(\Omega, \mathbb{R}^m)$ where the integrand
${\rm F}: \mathbb M_{m\times n}\to \mathbb{R}$ is a smooth uniformly
convex function
with bounded second derivatives. In this paper we address the
question of
regularity for solutions of the corresponding system of
EulerLagrange equations.
In particular we introduce a class of singular maps referred
to as traceless and
examine them as a new counterexample to the regularity of minimizers
of the energy
functional $\mathcal F[\cdot,\Omega]$ using a method based on
null Lagrangians.
Keywords:traceless map, singular minimizer, nullLagrangian Categories:49K27, 49N60, 49J30, 49K20 

5. CMB 2016 (vol 59 pp. 606)
 Mihăilescu, Mihai; Moroşanu, Gheorghe

Eigenvalues of $ \Delta_p \Delta_q $ Under Neumann Boundary Condition
The
eigenvalue problem $\Delta_p u\Delta_q u=\lambdau^{q2}u$
with $p\in(1,\infty)$, $q\in(2,\infty)$, $p\neq q$ subject to
the
corresponding homogeneous Neumann boundary condition is
investigated on a bounded open set with smooth boundary from
$\mathbb{R}^N$ with $N\geq 2$. A careful analysis of this problem leads
us to a complete description of the set of eigenvalues as being
a
precise interval $(\lambda_1, +\infty )$ plus an isolated point
$\lambda =0$. This comprehensive result is strongly related to
our
framework which is complementary to the wellknown case $p=q\neq
2$ for which a full description of the set of eigenvalues is
still
unavailable.
Keywords:eigenvalue problem, Sobolev space, Nehari manifold, variational methods Categories:35J60, 35J92, 46E30, 49R05 

6. CMB 2016 (vol 59 pp. 381)
 Moameni, Abbas

Supports of Extremal Doubly Stochastic Measures
A doubly stochastic measure on the unit square is a Borel probability
measure whose horizontal and vertical marginals both coincide
with the Lebesgue measure. The set of doubly stochastic measures
is convex and compact so its
extremal points are of particular interest. The problem number 111
of
Birkhoff (Lattice Theory 1948) is to provide a necessary and
sufficient condition on the support of a doubly stochastic measure
to guarantee extremality. It was proved by
BeneÅ¡ and Å tÄpÃ¡n that an extremal doubly stochastic measure is concentrated
on a set which admits an aperiodic decomposition.
Hestir and Williams later found a necessary condition which
is nearly sufficient by
further refining the aperiodic structure of the support of extremal
doubly stochastic measures.
Our objective in this work is to
provide a more practical necessary and nearly sufficient
condition for a set to support an extremal doubly stochastic
measure.
Keywords:optimal mass transport, doubly stochastic measures, extremality, uniqueness Category:49Q15 

7. CMB 2014 (vol 58 pp. 44)
8. CMB 2012 (vol 56 pp. 621)
 Shang, Yilun

Optimal Control Strategies for Virus Spreading in Inhomogeneous Epidemic Dynamics
In this paper, we study the spread of virus/worm in computer
networks with a view to addressing cyber security problems. Epidemic
models have been applied extensively to model the propagation of
computer viruses, which characterize the fact that infected machines
may spread malware to other hosts connected to the network. In our
framework, the dynamics of hosts evolves according to a modified
inhomogeneous SusceptibleInfectiousSusceptible (SIS) epidemic
model with timevarying transmission rate and recovery rate. The
infection of computers is subject to direct attack as well as
propagation among hosts. Based on optimal control theory, optimal
attack strategies are provided by minimizing the cost (equivalently
maximizing the profit) of the attacker. We present a threshold
function of the fraction of infectious hosts, which captures the
dynamically evolving strategies of the attacker and reflects the
persistence of virus spreading. Moreover, our results indicate that
if the infectivity of a computer worm is low and the computers are
installed with antivirus software with high reliability, the
intensity of attacks incurred will likely be low. This agrees with
our intuition.
Keywords:network securitypidemic dynamics, optimal control Categories:49J15, 92D30 

9. CMB 2011 (vol 56 pp. 272)
 Cheng, Lixin; Luo, Zhenghua; Zhou, Yu

On Super Weakly Compact Convex Sets and Representation of the Dual of the Normed Semigroup They Generate
In this note, we first give a characterization of super weakly
compact convex sets of a Banach space $X$:
a closed bounded convex set $K\subset X$ is
super weakly compact if and only if there exists a $w^*$ lower
semicontinuous seminorm $p$ with $p\geq\sigma_K\equiv\sup_{x\in
K}\langle\,\cdot\,,x\rangle$ such that $p^2$ is uniformly FrÃ©chet
differentiable on each bounded set of $X^*$. Then we present a
representation theorem for the dual of the semigroup $\textrm{swcc}(X)$
consisting of all the nonempty super weakly compact convex sets of the
space $X$.
Keywords:super weakly compact set, dual of normed semigroup, uniform FrÃ©chet differentiability, representation Categories:20M30, 46B10, 46B20, 46E15, 46J10, 49J50 

10. CMB 2011 (vol 55 pp. 697)
 Borwein, Jonathan M.; Vanderwerff, Jon

Constructions of Uniformly Convex Functions
We give precise conditions under which the composition
of a norm with a convex function yields a
uniformly convex function on a Banach space.
Various applications are given to functions of power type.
The results are dualized to study uniform smoothness
and several examples are provided.
Keywords:convex function, uniformly convex function, uniformly smooth function, power type, Fenchel conjugate, composition, norm Categories:52A41, 46G05, 46N10, 49J50, 90C25 

11. CMB 2011 (vol 55 pp. 723)
 Gigli, Nicola; Ohta, ShinIchi

First Variation Formula in Wasserstein Spaces over Compact Alexandrov Spaces
We extend results proved by the second author (Amer. J. Math., 2009)
for nonnegatively curved Alexandrov spaces
to general compact Alexandrov spaces $X$ with curvature bounded
below.
The gradient flow of a geodesically convex functional on the quadratic Wasserstein
space $(\mathcal P(X),W_2)$ satisfies the evolution variational inequality.
Moreover, the gradient flow enjoys uniqueness and contractivity.
These results are obtained by proving a first variation formula for
the Wasserstein distance.
Keywords:Alexandrov spaces, Wasserstein spaces, first variation formula, gradient flow Categories:53C23, 28A35, 49Q20, 58A35 

12. CMB 2009 (vol 40 pp. 88)
 Radulescu, M. L.; Clarke, F. H.

The multidirectional mean value theorem in Banach spaces
Recently, F.~H.~Clarke and Y.~Ledyaev established a
multidirectional mean value theorem applicable to lower
semicontinuous functions on Hilbert spaces, a result which
turns out to be useful in many applications. We develop a
variant of the result applicable to locally Lipschitz functions
on certain Banach spaces, namely those that admit a
${\cal C}^1$Lipschitz continuous bump function.
Categories:26B05, 49J52 

13. CMB 2005 (vol 48 pp. 283)
 Thibault, Lionel; Zagrodny, Dariusz

Enlarged Inclusion of Subdifferentials
This paper studies the integration of inclusion of subdifferentials. Under
various verifiable conditions, we obtain that if two proper lower
semicontinuous functions $f$ and $g$ have the subdifferential of $f$
included in the $\gamma$enlargement of the subdifferential of $g$, then
the difference of those functions is $ \gamma$Lipschitz over their
effective domain.
Keywords:subdifferential,, directionally regular function,, approximate convex function,, subdifferentially and directionally stable function Categories:49J52, 46N10, 58C20 

14. CMB 2002 (vol 45 pp. 154)
 Weitsman, Allen

On the Poisson Integral of Step Functions and Minimal Surfaces
Applications of minimal surface methods are made to obtain information
about univalent harmonic mappings. In the case where the mapping arises
as the Poisson integral of a step function, lower bounds for the number
of zeros of the dilatation are obtained in terms of the geometry of the
image.
Keywords:harmonic mappings, dilatation, minimal surfaces Categories:30C62, 31A05, 31A20, 49Q05 

15. CMB 2000 (vol 43 pp. 25)
 Bounkhel, M.; Thibault, L.

Subdifferential Regularity of Directionally Lipschitzian Functions
Formulas for the Clarke subdifferential are always expressed in the
form of inclusion. The equality form in these formulas generally
requires the functions to be directionally regular. This paper
studies the directional regularity of the general class of
extendedrealvalued functions that are directionally Lipschitzian.
Connections with the concept of subdifferential regularity are also
established.
Keywords:subdifferential regularity, directional regularity, directionally Lipschitzian functions Categories:49J52, 58C20, 49J50, 90C26 

16. CMB 1998 (vol 41 pp. 497)
 Borwein, J. M.; Girgensohn, R.; Wang, Xianfu

On the construction of HÃ¶lder and Proximal Subderivatives
We construct Lipschitz functions such that for all $s>0$ they are
$s$H\"older, and so proximally, subdifferentiable only on dyadic
rationals and nowhere else. As applications we construct Lipschitz
functions with prescribed H\"older and approximate subderivatives.
Keywords:Lipschitz functions, HÃ¶lder subdifferential, proximal subdifferential, approximate subdifferential, symmetric subdifferential, HÃ¶lder smooth, dyadic rationals Categories:49J52, 26A16, 26A24 

17. CMB 1998 (vol 41 pp. 41)
 Giner, E.

On the Clarke subdifferential of an integral functional on $L_p$, $1\leq p < \infty$
Given an integral functional defined on $L_p$, $1 \leq p <\infty$,
under a growth condition we give an upper bound of the Clarke
directional derivative and we obtain a nice inclusion between the
Clarke subdifferential of the integral functional and the set of
selections of the subdifferential of the integrand.
Keywords:Integral functional, integrand, epiderivative Categories:28A25, 49J52, 46E30 
