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1. CJM Online first

Yuan, Rirong
 On a class of fully nonlinear elliptic equations containing gradient terms on compact Hermitian manifolds In this paper we study a class of second order fully nonlinear elliptic equations containing gradient terms on compact Hermitian manifolds and obtain a priori estimates under proper assumptions close to optimal. The analysis developed here should be useful to deal with other Hessian equations containing gradient terms in other contexts. Keywords:Sasakian manifold, Hermitian manifold, subsolution, extra concavity condition, fully nonlinear elliptic equation containing gradient term on complex manifoldCategories:35J15, 53C55, 53C25, 35J25

2. CJM Online first

Wang, Xing; Zhang, Chunjie
 Pointwise convergence of solutions to the SchrÃ¶dinger equation on manifolds Let $(M^n,g)$ be a Riemannian manifold without boundary. We study the amount of initial regularity is required so that the solution to free SchrÃ¶dinger equation converges pointwise to its initial data. Assume the initial data is in $H^\alpha(M)$. For Hyperbolic Space, standard Sphere and the 2 dimensional Torus, we prove that $\alpha\gt \frac{1}{2}$ is enough. For general compact manifolds, due to lacking of local smoothing effect, it is hard to beat the bound $\alpha\gt 1$ from interpolation. We managed to go below 1 for dimension $\leq 3$. The more interesting thing is that, for 1 dimensional compact manifold, $\alpha\gt \frac{1}{3}$ is sufficient. Keywords:pointwise convergence, SchrÃ¶dinger operator, manifold, Strichartz estimateCategories:35L05, 46E35, 42B37

3. CJM Online first

Phan, Tuoc
 Lorentz estimates for weak solutions of quasi-linear parabolic equations with singular divergence-free drifts This paper investigates regularity in Lorentz spaces of weak solutions of a class of divergence form quasi-linear parabolic equations with singular divergence-free drifts. In this class of equations, the principal terms are vector field functions which are measurable in $(x,t)$-variable, and nonlinearly dependent on both unknown solutions and their gradients. Interior, local boundary, and global regularity estimates in Lorentz spaces for gradients of weak solutions are established assuming that the solutions are in BMO space, the John Nirenberg space. The results are even new when the drifts are identically zero because they do not require solutions to be bounded as in the available literature. In the linear setting, the results of the paper also improve the standard CalderÃ³n-Zygmund regularity theory to the critical borderline case. When the principal term in the equation does not depend on the solution as its variable, our results recover and sharpen known, available results. The approach is based on the perturbation technique introduced by Caffarelli and Peral together with a "double-scaling parameter" technique, and the maximal function free approach introduced by Acerbi and Mingione. Keywords:gradient estimate, quasi-linear parabolic equation, divergence-free driftCategories:35B45, 35K57, 35K59, 35K61

4. CJM 2016 (vol 69 pp. 854)

Saanouni, Tarek
 Global and non Global Solutions for Some Fractional Heat Equations with Pure Power Nonlinearity The initial value problem for a semi-linear fractional heat equation is investigated. In the focusing case, global well-posedness and exponential decay are obtained. In the focusing sign, global and non global existence of solutions are discussed via the potential well method. Keywords:nonlinear fractional heat equation, global Existence, decay, blow-upCategory:35Q55

5. CJM 2016 (vol 68 pp. 1334)

Jiang, Feida; Trudinger, Neil S; Xiang, Ni
 On the Neumann Problem for Monge-AmpÃ¨re Type Equations In this paper, we study the global regularity for regular Monge-AmpÃ¨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 Monge-AmpÃ¨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, Monge-AmpÃ¨re type equation, second derivative estimatesCategories:35J66, 35J96

6. CJM 2016 (vol 68 pp. 521)

Emamizadeh, Behrouz; Farjudian, Amin; Zivari-Rezapour, 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 conditionCategories:35J20, 35J25

7. CJM 2015 (vol 67 pp. 1161)

Zhang, Junqiang; Cao, Jun; Jiang, Renjin; Yang, Dachun
 Non-tangential 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 non-tangential 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 transformCategories:42B30, 42B35, 35J70

8. 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 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 SchrÃ¶dinger propagator, dispersive estimate, Dirichlet boundary condition, scattering theory, energy criticalCategories:35P25, 35Q55, 47J35

9. 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 HÃ¶lder inequalitiesCategory:35K20

10. 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, capacityCategories:35K08, 28A80, 31B05, 35J08, 46E35, 47D07

11. 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, CalderÃ³n-Zygmund operatorsCategories:30C62, 35J99, 42B20

12. 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 super-exponentiellement 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 demi-plan 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©sonancesCategories:35B34, 35P25

13. 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 reductionCategories:35J25, 35J65, 35J67

14. 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 solutionsCategories:35A01, 35A02, 35D30, 35J70, 35H20

15. 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, 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 spaceCategories:35A05, 76D03, 35B40, 35B10, 86A05, 86A10

16. 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 PDEsCategories:53Z05, 35Q99

17. 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, surfacesCategory:35R03

18. 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 domainCategories:35J25, 35J65

19. 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 solutionsCategories:35F60, 35F21, 35D40

20. 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

21. 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

22. 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 SchrÃ¶dinger 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 tunnelingCategories:37K40, 35Q55, 35Q51

23. 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 transformCategory:35P25

24. 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 identitiesCategories:28A80, , , , 35P20, 35J05, 43A05, 47A75

25. 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
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