1. CJM Online first
 Emamizadeh, Behrouz; Farjudian, Amin; ZivariRezapour, Mohsen

Optimization related to some nonlocal problems of Kirchhoff type
In this paper we introduce two rearrangement optimization
problems, one being a maximization and the other a minimization
problem, related to a nonlocal boundary value problem of Kirchhoff
type. Using the theory of rearrangements as developed by
G. R. Burton we are able to show that both problems are solvable,
and derive the corresponding optimality conditions. These conditions
in turn provide information concerning the locations of the
optimal
solutions. The strict convexity of the energy functional plays
a
crucial role in both problems. The popular case in which the
rearrangement class (i.e., the admissible set) is generated
by a
characteristic function is also considered. We show that in
this
case, the maximization problem gives rise to a free boundary
problem
of obstacle type, which turns out to be unstable. On the other
hand,
the minimization problem leads to another free boundary problem
of
obstacle type, which is stable. Some numerical results are
included
to confirm the theory.
Keywords:Kirchhoff equation, rearrangements of functions, maximization, existence, optimality condition Categories:35J20, 35J25 

2. CJM 2012 (vol 65 pp. 927)
 Wang, Liping; Zhao, Chunyi

Infinitely Many Solutions for the Prescribed Boundary Mean Curvature Problem in $\mathbb B^N$
We consider the following prescribed boundary mean curvature problem
in $ \mathbb B^N$ with the Euclidean metric:
\[
\begin{cases}
\displaystyle \Delta u =0,\quad u\gt 0 &\text{in }\mathbb B^N,
\\[2ex]
\displaystyle \frac{\partial u}{\partial\nu} + \frac{N2}{2} u =\frac{N2}{2} \widetilde K(x) u^{2^\#1} \quad & \text{on }\mathbb S^{N1},
\end{cases}
\]
where $\widetilde K(x)$ is positive and rotationally symmetric on $\mathbb
S^{N1}, 2^\#=\frac{2(N1)}{N2}$.
We show that if $\widetilde K(x)$ has a local maximum point,
then the above problem has infinitely many positive solutions
that are not rotationally symmetric on $\mathbb S^{N1}$.
Keywords:infinitely many solutions, prescribed boundary mean curvature, variational reduction Categories:35J25, 35J65, 35J67 

3. CJM 2012 (vol 65 pp. 702)
 Taylor, Michael

Regularity of Standing Waves on Lipschitz Domains
We analyze the regularity of standing wave solutions
to nonlinear SchrÃ¶dinger equations of power type on bounded domains,
concentrating on Lipschitz domains. We establish optimal regularity results
in this setting, in Besov spaces and in HÃ¶lder spaces.
Keywords:standing waves, elliptic regularity, Lipschitz domain Categories:35J25, 35J65 

4. CJM 2010 (vol 62 pp. 808)
 Legendre, Eveline

Extrema of Low Eigenvalues of the DirichletNeumann Laplacian on a Disk
We study extrema of the first and the second mixed eigenvalues of the Laplacian on the disk among some families of DirichletNeumann boundary conditions. We show that the minimizer of the second eigenvalue among all mixed boundary conditions lies in a compact $1$parameter family for which an explicit description is given. Moreover, we prove that among all partitions of the boundary with bounded number of parts on which Dirichlet and Neumann conditions are imposed alternately, the first eigenvalue is maximized by the uniformly distributed partition.
Keywords: Laplacian, eigenvalues, DirichletNeumann mixed boundary condition, Zaremba's problem Categories:35J25, 35P15 
