1. CJM Online first
2. CJM 2016 (vol 68 pp. 1334)
 Jiang, Feida; Trudinger, Neil S; Xiang, Ni

On the Neumann Problem for MongeAmpÃ¨re Type Equations
In this paper, we study the global regularity for
regular
MongeAmpÃ¨re type equations associated with semilinear Neumann
boundary conditions.
By establishing a priori estimates for second order derivatives,
the
classical solvability of the Neumann boundary value problem is
proved under natural conditions.
The techniques build upon the delicate and intricate treatment
of the standard MongeAmpÃ¨re case
by Lions, Trudinger and Urbas in 1986 and the recent barrier
constructions and second derivative bounds
by Jiang, Trudinger and Yang for the Dirichlet problem. We also
consider more general oblique boundary
value problems in the strictly regular case.
Keywords:semilinear Neumann problem, MongeAmpÃ¨re type equation, second derivative estimates Categories:35J66, 35J96 

3. CJM 2016 (vol 68 pp. 521)
 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 

4. CJM 2015 (vol 67 pp. 1161)
 Zhang, Junqiang; Cao, Jun; Jiang, Renjin; Yang, Dachun

Nontangential Maximal Function Characterizations of Hardy Spaces Associated with Degenerate Elliptic Operators
Let $w$ be either in the Muckenhoupt class of $A_2(\mathbb{R}^n)$ weights
or in the class of $QC(\mathbb{R}^n)$ weights, and
$L_w:=w^{1}\mathop{\mathrm{div}}(A\nabla)$
the degenerate elliptic operator on the Euclidean space $\mathbb{R}^n$,
$n\ge 2$. In this article, the authors establish the nontangential
maximal function characterization
of the Hardy space $H_{L_w}^p(\mathbb{R}^n)$ associated with $L_w$ for
$p\in (0,1]$ and, when $p\in (\frac{n}{n+1},1]$ and
$w\in A_{q_0}(\mathbb{R}^n)$ with $q_0\in[1,\frac{p(n+1)}n)$,
the authors prove that the associated Riesz transform $\nabla L_w^{1/2}$
is bounded from $H_{L_w}^p(\mathbb{R}^n)$ to the weighted classical
Hardy space $H_w^p(\mathbb{R}^n)$.
Keywords:degenerate elliptic operator, Hardy space, square function, maximal function, molecule, Riesz transform Categories:42B30, 42B35, 35J70 

5. CJM 2014 (vol 66 pp. 1110)
 Li, Dong; Xu, Guixiang; Zhang, Xiaoyi

On the Dispersive Estimate for the Dirichlet SchrÃ¶dinger Propagator and Applications to Energy Critical NLS
We consider the obstacle problem for the SchrÃ¶dinger evolution
in the exterior of the unit ball with Dirichlet boundary condition. Under
the radial symmetry we compute explicitly the fundamental solution
for the linear Dirichlet SchrÃ¶dinger
propagator $e^{it\Delta_D}$
and give a robust algorithm to prove sharp $L^1 \rightarrow
L^{\infty}$ dispersive estimates. We showcase the analysis in
dimensions $n=5,7$. As an application, we obtain global
wellposedness and scattering for defocusing energycritical NLS on
$\Omega=\mathbb{R}^n\backslash \overline{B(0,1)}$ with Dirichlet boundary
condition and radial data in these dimensions.
Keywords:Dirichlet SchrÃ¶dinger propagator, dispersive estimate, Dirichlet boundary condition, scattering theory, energy critical Categories:35P25, 35Q55, 47J35 

6. CJM 2013 (vol 66 pp. 429)
 RiveraNoriega, Jorge

Perturbation and Solvability of Initial $L^p$ Dirichlet Problems for Parabolic Equations over Noncylindrical Domains
For parabolic linear operators $L$ of second order in divergence form,
we prove that the solvability of initial $L^p$ Dirichlet problems for
the whole range $1\lt p\lt \infty$ is preserved under appropriate small
perturbations of the coefficients of the operators involved.
We also prove that if the coefficients of $L$ satisfy a suitable
controlled oscillation in the form of Carleson measure conditions,
then for certain values of $p\gt 1$, the initial $L^p$ Dirichlet problem
associated to $Lu=0$ over noncylindrical domains is solvable.
The results are adequate adaptations of the corresponding results for
elliptic equations.
Keywords:initial $L^p$ Dirichlet problem, second order parabolic equations in divergence form, noncylindrical domains, reverse HÃ¶lder inequalities Category:35K20 

7. CJM 2013 (vol 66 pp. 641)
 Grigor'yan, Alexander; Hu, Jiaxin

Heat Kernels and Green Functions on Metric Measure Spaces
We prove that, in a setting of local Dirichlet forms on metric measure
spaces, a twosided subGaussian estimate of the heat kernel is equivalent
to the conjunction of the volume doubling propety, the elliptic Harnack
inequality and a certain estimate of the capacity between concentric balls.
The main technical tool is the equivalence between the capacity estimate and
the estimate of a mean exit time in a ball, that uses twosided estimates of
a Green function in a ball.
Keywords:Dirichlet form, heat kernel, Green function, capacity Categories:35K08, 28A80, 31B05, 35J08, 46E35, 47D07 

8. CJM 2013 (vol 65 pp. 1217)
 Cruz, Victor; Mateu, Joan; Orobitg, Joan

Beltrami Equation with Coefficient in Sobolev and Besov Spaces
Our goal in this work is to present some function spaces on the
complex plane $\mathbb C$, $X(\mathbb C)$, for which the quasiregular solutions of
the Beltrami equation, $\overline\partial f (z) = \mu(z) \partial f
(z)$, have first derivatives locally in $X(\mathbb C)$, provided that the
Beltrami coefficient $\mu$ belongs to $X(\mathbb C)$.
Keywords:quasiregular mappings, Beltrami equation, Sobolev spaces, CalderÃ³nZygmund operators Categories:30C62, 35J99, 42B20 

9. CJM 2013 (vol 65 pp. 1095)
 Sambou, Diomba

RÃ©sonances prÃ¨s de seuils d'opÃ©rateurs magnÃ©tiques de Pauli et de Dirac
Nous considÃ©rons les perturbations $H := H_{0} + V$ et $D := D_{0} +
V$ des Hamiltoniens libres $H_{0}$ de Pauli et $D_{0}$ de Dirac en
dimension 3 avec champ magnÃ©tique non constant, $V$ Ã©tant un
potentiel Ã©lectrique qui dÃ©croÃ®t superexponentiellement dans la
direction du champ magnÃ©tique. Nous montrons que dans des espaces de
Banach appropriÃ©s, les rÃ©solvantes de $H$ et $D$ dÃ©finies sur le
demiplan supÃ©rieur admettent des prolongements mÃ©romorphes. Nous
dÃ©finissons les rÃ©sonances de $H$ et $D$ comme Ã©tant les pÃ´les de
ces extensions mÃ©romorphes. D'une part, nous Ã©tudions la
rÃ©partition des rÃ©sonances de $H$ prÃ¨s de l'origine $0$ et d'autre
part, celle des rÃ©sonances de $D$ prÃ¨s de $\pm m$ oÃ¹ $m$ est la
masse d'une particule. Dans les deux cas, nous obtenons d'abord des
majorations du nombre de rÃ©sonances dans de petits domaines au
voisinage de $0$ et $\pm m$. Sous des hypothÃ¨ses supplÃ©mentaires,
nous obtenons des dÃ©veloppements asymptotiques du nombre de
rÃ©sonances qui entraÃ®nent leur accumulation prÃ¨s des seuils $0$ et
$\pm m$. En particulier, pour une perturbation $V$ de signe dÃ©fini,
nous obtenons des informations sur la rÃ©partition des valeurs propres
de $H$ et $D$ prÃ¨s de $0$ et $\pm m$ respectivement.
Keywords:opÃ©rateurs magnÃ©tiques de Pauli et de Dirac, rÃ©sonances Categories:35B34, 35P25 

10. 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 

11. CJM 2012 (vol 64 pp. 1395)
 Rodney, Scott

Existence of Weak Solutions of Linear Subelliptic Dirichlet Problems With Rough Coefficients
This article gives an existence theory for weak solutions of second order nonelliptic linear Dirichlet problems of the form
\begin{align*}
\nabla'P(x)\nabla u +{\bf HR}u+{\bf S'G}u +Fu &= f+{\bf T'g} \text{ in }\Theta
\\
u&=\varphi\text{ on }\partial \Theta.
\end{align*}
The principal part $\xi'P(x)\xi$ of the above equation is assumed to
be comparable to a quadratic form ${\mathcal Q}(x,\xi) = \xi'Q(x)\xi$ that
may vanish for nonzero $\xi\in\mathbb{R}^n$. This is achieved using
techniques of functional analysis applied to the degenerate Sobolev
spaces $QH^1(\Theta)=W^{1,2}(\Theta,Q)$ and
$QH^1_0(\Theta)=W^{1,2}_0(\Theta,Q)$ as defined in
previous works.
Sawyer and Wheeden give a regularity theory
for a subset of the class of equations dealt with here.
Keywords:degenerate quadratic forms, linear equations, rough coefficients, subelliptic, weak solutions Categories:35A01, 35A02, 35D30, 35J70, 35H20 

12. CJM 2012 (vol 64 pp. 1415)
 Selmi, Ridha

Global WellPosedness and Convergence Results for 3DRegularized Boussinesq System
Analytical study to the regularization of the Boussinesq system is
performed in frequency space using Fourier theory. Existence and
uniqueness of weak solution with minimum regularity requirement are
proved. Convergence results of the unique weak solution of the
regularized Boussinesq system to a weak LerayHopf solution of the
Boussinesq system are established as the regularizing parameter
$\alpha$ vanishes. The proofs are done in the frequency space and use
energy methods, ArselÃ Ascoli compactness theorem and a Friedrichs
like approximation scheme.
Keywords:regularizing Boussinesq system, existence and uniqueness of weak solution, convergence results, compactness method in frequency space Categories:35A05, 76D03, 35B40, 35B10, 86A05, 86A10 

13. CJM 2012 (vol 65 pp. 655)
 Shemyakova, E.

Proof of the Completeness of Darboux Wronskian Formulae for Order Two
Darboux Wronskian formulas allow to construct Darboux transformations,
but Laplace transformations, which are Darboux transformations of
order one
cannot be represented this way.
It has been a long standing problem on what are other exceptions. In
our previous work we proved that among transformations of total
order one there are no other exceptions. Here we prove that for
transformations of total order two there are no exceptions at all.
We also obtain a simple explicit invariant description of all possible
Darboux Transformations of total order two.
Keywords:completeness of Darboux Wronskian formulas, completeness of Darboux determinants, Darboux transformations, invariants for solution of PDEs Categories:53Z05, 35Q99 

14. CJM 2012 (vol 65 pp. 621)
 Lee, Paul W. Y.

On Surfaces in Three Dimensional Contact Manifolds
In this paper, we introduce two notions on a surface in a contact
manifold. The first one is called degree of transversality (DOT) which
measures the transversality between the tangent spaces of a surface
and the contact planes. The second quantity, called curvature of
transversality (COT), is designed to give a comparison principle for
DOT along characteristic curves under bounds on COT. In particular,
this gives estimates on lengths of characteristic curves assuming COT
is bounded below by a positive constant.
We show that surfaces with constant COT exist and we classify all
graphs in the Heisenberg group with vanishing COT. This is
accomplished by showing that the equation for graphs with zero COT can
be decomposed into two first order PDEs, one of which is the backward
invisicid Burgers' equation. Finally we show that the pminimal graph
equation in the Heisenberg group also has such a
decomposition. Moreover, we can use this decomposition to write down
an explicit formula of a solution near a regular point.
Keywords:contact manifolds, subriemannian manifolds, surfaces Category:35R03 

15. 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 

16. CJM 2011 (vol 64 pp. 1289)
 Gomes, Diogo; Serra, António

Systems of Weakly Coupled HamiltonJacobi Equations with Implicit Obstacles
In this paper we study systems of weakly coupled HamiltonJacobi equations
with implicit obstacles that arise in optimal switching problems.
Inspired by methods from the theory of viscosity solutions and
weak KAM theory, we
extend the notion of Aubry set for these
systems. This enables us
to prove a new result on existence and uniqueness of
solutions for the Dirichlet problem, answering a question
of F. Camilli, P. Loreti and N. Yamada.
Keywords:HamiltonJacobi equations, switching costs, viscosity solutions Categories:35F60, 35F21, 35D40 

17. CJM 2011 (vol 64 pp. 924)
 McCann, Robert J.; Pass, Brendan; Warren, Micah

Rectifiability of Optimal Transportation Plans
The regularity of solutions to optimal transportation problems has become
a hot topic in current research. It is well known by now that the optimal measure
may not be concentrated on the graph of a continuous mapping unless both the transportation
cost and the masses transported satisfy very restrictive hypotheses (including sign conditions
on the mixed fourthorder derivatives of the cost function).
The purpose of this note is to show that in spite of this,
the optimal measure is supported on a Lipschitz manifold, provided only
that the cost is $C^{2}$ with nonsingular mixed second derivative.
We use this result to provide a simple proof that solutions to Monge's
optimal transportation problem satisfy a change of variables equation
almost everywhere.
Categories:49K20, 49K60, 35J96, 58C07 

18. CJM 2011 (vol 64 pp. 217)
19. CJM 2011 (vol 63 pp. 1201)
20. CJM 2011 (vol 63 pp. 961)
 Bouclet, JeanMarc

Low Frequency Estimates for Long Range Perturbations in Divergence Form
We prove a uniform control as $ z \rightarrow 0 $ for the resolvent $
(Pz)^{1} $ of long range perturbations $ P $ of the Euclidean
Laplacian in divergence form by combining positive commutator
estimates and properties of Riesz transforms. These estimates hold in
dimension $d \geq 3 $ when $ P $ is defined on $ \mathbb{R}^d $ and in dimension $ d \geq 2 $ when $ P $ is defined outside a compact obstacle with Dirichlet boundary conditions.
Keywords:resolvent estimates, thresholds, scattering theory, Riesz transform Category:35P25 

21. CJM 2011 (vol 63 pp. 648)
 Ngai, SzeMan

Spectral Asymptotics of Laplacians Associated with Onedimensional Iterated Function Systems with Overlaps
We set up a framework for computing the spectral dimension of a class of onedimensional
selfsimilar measures that are defined by iterated function systems
with overlaps and satisfy a family of secondorder selfsimilar
identities. As applications of our result we obtain the spectral dimension
of important measures such as the infinite Bernoulli convolution
associated with the golden ratio and convolutions of Cantortype measures.
The main novelty of our result is that the iterated function systems
we consider are not postcritically finite and do not satisfy the
wellknown open set condition.
Keywords:spectral dimension, fractal, Laplacian, selfsimilar measure, iterated function system with overlaps, secondorder selfsimilar identities Categories:28A80, , , , 35P20, 35J05, 43A05, 47A75 

22. CJM 2010 (vol 63 pp. 153)
 Hambly, B. M.

Asymptotics for Functions Associated with Heat Flow on the Sierpinski Carpet
We establish the asymptotic behaviour of the partition function, the
heat content, the integrated eigenvalue counting function, and, for
certain points, the ondiagonal heat kernel of generalized
Sierpinski carpets. For all these functions the leading term is of
the form $x^{\gamma}\phi(\log x)$ for a suitable exponent $\gamma$
and $\phi$ a periodic function. We also discuss similar results for
the heat content of affine nested fractals.
Categories:35K05, 28A80, 35B40, 60J65 

23. CJM 2010 (vol 63 pp. 55)
 Chau, Albert; Tam, LuenFai; Yu, Chengjie

Pseudolocality for the Ricci Flow and Applications
Perelman established a differential LiYauHamilton
(LYH) type inequality for fundamental solutions of the conjugate
heat equation corresponding to the Ricci flow on compact manifolds.
As an application of the LYH inequality,
Perelman proved a pseudolocality result for the Ricci flow on
compact manifolds. In this article we provide the details for the
proofs of these results in the case of a complete noncompact
Riemannian manifold. Using these results we prove that under
certain conditions, a finite time singularity of the Ricci flow
must form within a compact set. The conditions are satisfied by
asymptotically flat manifolds. We also prove a long time existence
result for the K\"ahlerRicci flow on complete nonnegatively curved K\"ahler
manifolds.
Categories:53C44, 58J37, 35B35 

24. 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 

25. CJM 2010 (vol 62 pp. 1116)
 Jin, Yongyang; Zhang, Genkai

Degenerate pLaplacian Operators and Hardy Type Inequalities on
HType Groups
Let $\mathbb G$ be a steptwo nilpotent group of Htype with Lie algebra $\mathfrak G=V\oplus \mathfrak t$. We define a class of vector fields $X=\{X_j\}$ on $\mathbb G$ depending on a real parameter $k\ge 1$, and we consider the corresponding $p$Laplacian operator $L_{p,k} u= \operatorname{div}_X (\nabla_{X} u^{p2} \nabla_X u)$. For $k=1$ the vector fields $X=\{X_j\}$ are the left invariant vector fields corresponding to an orthonormal basis of $V$; for $\mathbb G$ being the Heisenberg group the vector fields are the Greiner fields. In this paper we obtain the fundamental solution for the operator $L_{p,k}$ and as an application, we get a Hardy type inequality associated with $X$.
Keywords:fundamental solutions, degenerate Laplacians, Hardy inequality, Htype groups Categories:35H30, 26D10, 22E25 
