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1. CMB Online first

Steinerberger, Stefan
An Endpoint Alexandrov Bakelman Pucci estimate in the Plane
The classical Alexandrov-Bakelman-Pucci 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 $$ \max_{x \in \Omega}{ |u(x)|} \leq \max_{x \in \partial \Omega}{|u(x)|} + c \max_{x \in \Omega} \int_{y \in \Omega}{ \max \left\{ 1, \log{ \left(\frac{|\Omega|}{\|x-y\|^2} \right)} \right\} \left| \Delta u(y) \right| dy}.$$ This is somewhat dual to the classical Trudinger-Moser inequality; we also note that it is sharper than the usual estimates given in Orlicz spaces, the proof is rearrangement-free. The Laplacian can be replaced by any uniformly elliptic operator in divergence form.

Keywords:Alexandrov-Bakelman-Pucci estimate, second order Sobolev inequality, Trudinger-Moser inequality
Categories:35A23, 35B50, 28A75, 49Q20

2. CMB Online first

Haslhofer, Robert; Ivaki, Mohammad N.
Low complexity solutions of the Allen-Cahn equation on three-spheres
In this short note, we prove that on the three-sphere with any bumpy metric there exist at least two pairs of solutions of the Allen-Cahn equation with spherical interface and index at most two. The proof combines several recent results from the literature.

Keywords:Allen-Cahn equation, phase transition, small index

3. CMB 2017 (vol 61 pp. 495)

Chang, Der-Chen; Yang, Nanping; Wu, Hsi-Chun
Poincaré Lemma on Quaternion-Like 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, Poincaré lemma
Categories:93B05, 49N99

4. CMB 2017 (vol 60 pp. 631)

Shahrokhi-Dehkordi, M. S.
Traceless Maps as the Singular Minimizers in the Multi-dimensional 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 Euler-Lagrange 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, null-Lagrangian
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 $-\Delta_p u-\Delta_q u=\lambda|u|^{q-2}u$ 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 well-known 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 Beneš and Štėpán 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

7. CMB 2014 (vol 58 pp. 44)

Daniilidis, A.; Drusvyatskiy, D.; Lewis, A. S.
Orbits of Geometric Descent
We prove that quasiconvex functions always admit descent trajectories bypassing all non-minimizing critical points.

Keywords:differential inclusion, quasiconvex function, self-contracted curve, sweeping process
Categories:34A60, 49J99

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 Susceptible-Infectious-Susceptible (SIS) epidemic model with time-varying 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 Fréchet 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 Fréchet 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, Shin-Ichi
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 semi-continuous 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 extended-real-valued 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 Hölder 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, Hölder subdifferential, proximal subdifferential, approximate subdifferential, symmetric subdifferential, Hölder 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, epi-derivative
Categories:28A25, 49J52, 46E30

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