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1. CJM Online first

Li, Dong; Xu, Guixiang; Zhang, Xiaoyi
On the Dispersive Estimate for the Dirichlet Schrödinger Propagator and Applications to Energy Critical NLS
We consider the obstacle problem for the Schrödinger 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 Schrödinger 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 well-posedness and scattering for defocusing energy-critical NLS on $\Omega=\mathbb{R}^n\backslash \overline{B(0,1)}$ with Dirichlet boundary condition and radial data in these dimensions.

Keywords:Dirichlet Schrödinger propagator, dispersive estimate, Dirichlet boundary condition, scattering theory, energy critical
Categories:35P25, 35Q55, 47J35

2. CJM 2013 (vol 66 pp. 429)

Rivera-Noriega, Jorge
Perturbation and Solvability of Initial $L^p$ Dirichlet Problems for Parabolic Equations over Non-cylindrical 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 non-cylindrical 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, non-cylindrical domains, reverse Hölder inequalities
Category:35K20

3. 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, Calderón-Zygmund operators
Categories:30C62, 35J99, 42B20

4. 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 two-sided sub-Gaussian 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 two-sided estimates of a Green function in a ball.

Keywords:Dirichlet form, heat kernel, Green function, capacity
Categories:35K08, 28A80, 31B05, 35J08, 46E35, 47D07

5. CJM 2013 (vol 65 pp. 1095)

Sambou, Diomba
Résonances près de seuils d'opérateurs magnétiques de Pauli et de Dirac
Nous considérons 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 magnétique non constant, $V$ étant un potentiel électrique qui décroît super-exponentiellement dans la direction du champ magnétique. Nous montrons que dans des espaces de Banach appropriés, les résolvantes de $H$ et $D$ définies sur le demi-plan supérieur admettent des prolongements méromorphes. Nous définissons les résonances de $H$ et $D$ comme étant les pôles de ces extensions méromorphes. D'une part, nous étudions la répartition des résonances de $H$ près de l'origine $0$ et d'autre part, celle des résonances de $D$ près de $\pm m$ où $m$ est la masse d'une particule. Dans les deux cas, nous obtenons d'abord des majorations du nombre de résonances dans de petits domaines au voisinage de $0$ et $\pm m$. Sous des hypothèses supplémentaires, nous obtenons des développements asymptotiques du nombre de résonances qui entraînent leur accumulation près des seuils $0$ et $\pm m$. En particulier, pour une perturbation $V$ de signe défini, nous obtenons des informations sur la répartition des valeurs propres de $H$ et $D$ près de $0$ et $\pm m$ respectivement.

Keywords:opérateurs magnétiques de Pauli et de Dirac, résonances
Categories:35B34, 35P25

6. 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: \[ \begin{cases} \displaystyle -\Delta u =0,\quad u\gt 0 &\text{in }\mathbb B^N, \\[2ex] \displaystyle \frac{\partial u}{\partial\nu} + \frac{N-2}{2} u =\frac{N-2}{2} \widetilde K(x) u^{2^\#-1} \quad & \text{on }\mathbb S^{N-1}, \end{cases} \] where $\widetilde K(x)$ is positive and rotationally symmetric on $\mathbb S^{N-1}, 2^\#=\frac{2(N-1)}{N-2}$. 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 $\mathbb S^{N-1}$.

Keywords:infinitely many solutions, prescribed boundary mean curvature, variational reduction
Categories:35J25, 35J65, 35J67

7. 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 non-elliptic 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 non-zero $\xi\in\mathbb{R}^n$. 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

8. CJM 2012 (vol 64 pp. 1415)

Selmi, Ridha
Global Well-Posedness and Convergence Results for 3D-Regularized 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 Leray-Hopf 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, Arselà-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

9. 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

10. 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 p-minimal 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

11. 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 Schrödinger equations of power type on bounded domains, concentrating on Lipschitz domains. We establish optimal regularity results in this setting, in Besov spaces and in Hölder spaces.

Keywords:standing waves, elliptic regularity, Lipschitz domain
Categories:35J25, 35J65

12. CJM 2011 (vol 64 pp. 1289)

Gomes, Diogo; Serra, António
Systems of Weakly Coupled Hamilton-Jacobi Equations with Implicit Obstacles
In this paper we study systems of weakly coupled Hamilton-Jacobi 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:Hamilton-Jacobi equations, switching costs, viscosity solutions
Categories:35F60, 35F21, 35D40

13. 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 fourth-order 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 non-singular 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

14. CJM 2011 (vol 64 pp. 217)

Tang, Lin
$W_\omega^{2,p}$-Solvability of the Cauchy-Dirichlet Problem for Nondivergence Parabolic Equations with BMO Coefficients
In this paper, we establish the regularity of strong solutions to nondivergence parabolic equations with BMO coefficients in nondoubling weighted spaces.

Categories:35J45, 35J55

15. CJM 2011 (vol 63 pp. 1201)

Abou Salem, Walid K. ; Sulem, Catherine
Resonant Tunneling of Fast Solitons through Large Potential Barriers
We rigorously study the resonant tunneling of fast solitons through large potential barriers for the nonlinear Schrödinger equation in one dimension. Our approach covers the case of general nonlinearities, both local and Hartree (nonlocal).

Keywords:nonlinear Schroedinger equations, external potential, solitary waves, long time behavior, resonant tunneling
Categories:37K40, 35Q55, 35Q51

16. CJM 2011 (vol 63 pp. 961)

Bouclet, Jean-Marc
Low Frequency Estimates for Long Range Perturbations in Divergence Form
We prove a uniform control as $ z \rightarrow 0 $ for the resolvent $ (P-z)^{-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

17. CJM 2011 (vol 63 pp. 648)

Ngai, Sze-Man
Spectral Asymptotics of Laplacians Associated with One-dimensional Iterated Function Systems with Overlaps
We set up a framework for computing the spectral dimension of a class of one-dimensional self-similar measures that are defined by iterated function systems with overlaps and satisfy a family of second-order self-similar 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 Cantor-type measures. The main novelty of our result is that the iterated function systems we consider are not post-critically finite and do not satisfy the well-known open set condition.

Keywords:spectral dimension, fractal, Laplacian, self-similar measure, iterated function system with overlaps, second-order self-similar identities
Categories:28A80, , , , 35P20, 35J05, 43A05, 47A75

18. 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 on-diagonal 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

19. CJM 2010 (vol 63 pp. 55)

Chau, Albert; Tam, Luen-Fai; Yu, Chengjie
Pseudolocality for the Ricci Flow and Applications
Perelman established a differential Li--Yau--Hamilton (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\"ahler--Ricci flow on complete nonnegatively curved K\"ahler manifolds.

Categories:53C44, 58J37, 35B35

20. CJM 2010 (vol 62 pp. 808)

Legendre, Eveline
Extrema of Low Eigenvalues of the Dirichlet-Neumann 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 Dirichlet--Neumann 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, Dirichlet-Neumann mixed boundary condition, Zaremba's problem
Categories:35J25, 35P15

21. CJM 2010 (vol 62 pp. 1116)

Jin, Yongyang; Zhang, Genkai
Degenerate p-Laplacian Operators and Hardy Type Inequalities on H-Type Groups
Let $\mathbb G$ be a step-two nilpotent group of H-type 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 $L_{p,k} u= \operatorname{div}_X (|\nabla_{X} u|^{p-2} \nabla_X u)$. 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, H-type groups
Categories:35H30, 26D10, 22E25

22. CJM 2009 (vol 62 pp. 19)

Bouchekif, Mohammed; Nasri, Yasmina
Solutions for Semilinear Elliptic Systems with Critical Sobolev Exponent and Hardy Potential
In this paper we consider an elliptic system with an inverse square potential and critical Sobolev exponent in a bounded domain of $\mathbb{R}^N$. By variational methods we study the existence results.

Keywords:critical Sobolev exponent, Palais--Smale condition, Linking theorem, Hardy potential
Categories:35B25, 35B33, 35J50, 35J60

23. CJM 2009 (vol 62 pp. 202)

Tang, Lin
Interior $h^1$ Estimates for Parabolic Equations with $\operatorname{LMO}$ Coefficients
In this paper we establish \emph{a priori} $h^1$-estimates in a bounded domain for parabolic equations with vanishing $\operatorname{LMO}$ coefficients.

Keywords:parabolic operator, Hardy space, parabolic, singular integrals and commutators
Categories:35K20, 35B65, 35R05

24. CJM 2009 (vol 62 pp. 74)

Ducrot, Arnaud; Liu, Zhihua; Magal, Pierre
Projectors on the Generalized Eigenspaces for Neutral Functional Differential Equations in $L^{p}$ Spaces
We present the explicit formulas for the projectors on the generalized eigenspaces associated with some eigenvalues for linear neutral functional differential equations (NFDE) in $L^{p}$ spaces by using integrated semigroup theory. The analysis is based on the main result established elsewhere by the authors and results by Magal and Ruan on non-densely defined Cauchy problem. We formulate the NFDE as a non-densely defined Cauchy problem and obtain some spectral properties from which we then derive explicit formulas for the projectors on the generalized eigenspaces associated with some eigenvalues. Such explicit formulas are important in studying bifurcations in some semi-linear problems.

Keywords:neutral functional differential equations, semi-linear problem, integrated semigroup, spectrum, projectors
Categories:34K05, 35K57, 47A56, 47H20

25. CJM 2009 (vol 61 pp. 721)

Calin, Ovidiu; Chang, Der-Chen; Markina, Irina
SubRiemannian Geometry on the Sphere $\mathbb{S}^3$
We discuss the subRiemannian geometry induced by two noncommutative vector fields which are left invariant on the Lie group $\mathbb{S}^3$.

Keywords:noncommutative Lie group, quaternion group, subRiemannian geodesic, horizontal distribution, connectivity theorem, holonomic constraint
Categories:53C17, 53C22, 35H20
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