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1. CJM Online first
On the Dispersive Estimate for the Dirichlet SchrÃ¶dinger Propagator and Applications to Energy Critical NLS |
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 critical Categories:35P25, 35Q55, 47J35 |
2. CJM 2013 (vol 66 pp. 429)
Perturbation and Solvability of Initial $L^p$ Dirichlet Problems for Parabolic Equations over Non-cylindrical Domains |
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 inequalities Category:35K20 |
3. CJM 2013 (vol 65 pp. 1217)
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 operators Categories:30C62, 35J99, 42B20 |
4. CJM 2013 (vol 66 pp. 641)
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)
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©sonances Categories:35B34, 35P25 |
6. CJM 2012 (vol 65 pp. 927)
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)
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)
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 space Categories:35A05, 76D03, 35B40, 35B10, 86A05, 86A10 |
9. CJM 2012 (vol 65 pp. 655)
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)
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)
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 |
12. CJM 2011 (vol 64 pp. 1289)
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)
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)
$W_\omega^{2,p}$-Solvability of the Cauchy-Dirichlet Problem for Nondivergence Parabolic Equations with BMO Coefficients |
$W_\omega^{2,p}$-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)
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 tunneling Categories:37K40, 35Q55, 35Q51 |
16. CJM 2011 (vol 63 pp. 961)
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)
Spectral Asymptotics of Laplacians Associated with One-dimensional Iterated Function Systems with Overlaps |
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)
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)
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)
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)
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)
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)
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)
Projectors on the Generalized Eigenspaces for Neutral Functional Differential Equations in $L^{p}$ Spaces |
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)
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 |