Expand all Collapse all | Results 1 - 21 of 21 |
1. CJM 2013 (vol 65 pp. 740)
Regularization of Subsolutions in Discrete Weak KAM Theory We expose different methods of regularizations of subsolutions
in the context of discrete weak KAM theory.
They allow to prove the existence and the density of $C^{1,1}$
subsolutions. Moreover, these subsolutions can be made strict
and smooth outside of the Aubry set.
Keywords:discrete subsolutions, regularity Category:49C15 |
2. 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 |
3. 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 |
4. 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 |
5. 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 |
6. 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 |
7. CJM 2011 (vol 63 pp. 689)
Higher Rank Wavelets A theory of higher rank multiresolution analysis is given in the
setting of abelian multiscalings. This theory enables the
construction, from a higher rank MRA, of finite wavelet sets
whose multidilations have translates forming an orthonormal basis
in $L^2(\mathbb R^d)$. While tensor products of uniscaled MRAs provide
simple examples we construct many nonseparable higher rank
wavelets. In particular we construct \emph{Latin square
wavelets} as rank~$2$ variants of Haar wavelets. Also we construct
nonseparable scaling functions for rank $2$ variants of Meyer
wavelet scaling functions, and we construct the associated
nonseparable wavelets with compactly supported Fourier transforms.
On the other hand we show that compactly supported scaling
functions for biscaled MRAs are necessarily separable.
Keywords: wavelet, multi-scaling, higher rank, multiresolution, Latin squares Categories:42C40, 42A65, 42A16, 43A65 |
8. 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 |
9. CJM 2008 (vol 60 pp. 334)
Low-Pass Filters and Scaling Functions for Multivariable Wavelets We show that a characterization of scaling functions for
multiresolution analyses given by Hern\'{a}ndez and Weiss and that a
characterization of low-pass filters given by Gundy both hold for
multivariable multiresolution analyses.
Keywords:multivariable multiresolution analysis, low-pass filter, scaling function Categories:42C40, 60G35 |
10. CJM 2007 (vol 59 pp. 1135)
Sobolev Extensions of HÃ¶lder Continuous and Characteristic Functions on Metric Spaces We study when characteristic and H\"older continuous functions
are traces of Sobolev functions on doubling metric measure spaces.
We provide analytic and geometric conditions sufficient for extending
characteristic and H\"older continuous functions into globally defined
Sobolev functions.
Keywords:characteristic function, Newtonian function, metric space, resolutivity, HÃ¶lder continuous, Perron solution, $p$-harmonic, Sobolev extension, Whitney covering Categories:46E35, 31C45 |
11. CJM 2007 (vol 59 pp. 1301)
Strichartz Inequalities for the Wave Equation with the Full Laplacian on the Heisenberg Group We prove dispersive and Strichartz inequalities for the solution of the wave
equation related to the full
Laplacian on the Heisenberg group, by means of Besov spaces defined by a
Littlewood--Paley
decomposition related to the spectral resolution of the full Laplacian.
This requires a careful
analysis due also to the non-homogeneous nature of the full Laplacian.
This result has to be compared to a previous one by Bahouri, G\'erard
and Xu concerning the solution of the wave equation related to
the Kohn Laplacian.
Keywords:nilpotent and solvable Lie groups, smoothness and regularity of solutions of PDEs Categories:22E25, 35B65 |
12. CJM 2007 (vol 59 pp. 332)
Endomorphism Rings of Finite Global Dimension For a commutative local ring $R$, consider (noncommutative)
$R$-algebras $\Lambda$ of the form $\Lambda = \operatorname{End}_R(M)$
where $M$ is a reflexive $R$-module with nonzero free direct summand.
Such algebras $\Lambda$ of finite global dimension can be viewed as
potential substitutes for, or analogues of, a resolution of
singularities of $\operatorname{Spec} R$. For example, Van den Bergh
has shown that a three-dimensional Gorenstein normal
$\mathbb{C}$-algebra with isolated terminal singularities has a
crepant resolution of singularities if and only if it has such an
algebra $\Lambda$ with finite global dimension and which is maximal
Cohen--Macaulay over $R$ (a ``noncommutative crepant resolution of
singularities''). We produce algebras
$\Lambda=\operatorname{End}_R(M)$ having finite global dimension in
two contexts: when $R$ is a reduced one-dimensional complete local
ring, or when $R$ is a Cohen--Macaulay local ring of finite
Cohen--Macaulay type. If in the latter case $R$ is Gorenstein, then
the construction gives a noncommutative crepant resolution of
singularities in the sense of Van den Bergh.
Keywords:representation dimension, noncommutative crepant resolution, maximal Cohen--Macaulay modules Categories:16G50, 16G60, 16E99 |
13. CJM 2006 (vol 58 pp. 449)
Existence and Multiplicity of Positive Solutions for Singular Semipositone $p$-Laplacian Equations Positive solutions are obtained for the boundary value problem
\[\begin{cases}
-( | u'| ^{p-2}u')'
=\lambda f( t,u),\;t\in ( 0,1) ,p>1\\
u( 0) =u(1) =0.
\end{cases}
\]
Here $f(t,u) \geq -M,$ ($M$ is a positive constant)
for $(t,u) \in [0\mathinner{,}1] \times (0,\infty )$.
We will show the existence of two positive
solutions by using degree theory together with the upper-lower
solution method.
Keywords:one dimensional $p$-Laplacian, positive solution, degree theory, upper and lower solution Category:34B15 |
14. CJM 2005 (vol 57 pp. 61)
On Operators with Spectral Square but without Resolvent Points Decompositions of spectral type are
obtained for closed Hilbert space operators with empty resolvent
set, but whose square has closure which is spectral. Krein space
situations are also discussed.
Keywords:unbounded operators, closed operators,, spectral resolution, indefinite metric Categories:47A05, 47A15, 47B40, 47B50, 46C20 |
15. CJM 2002 (vol 54 pp. 1121)
Fully Nonlinear Elliptic Equations on General Domains By means of the Pucci operator, we construct a function $u_0$, which plays
an essential role in our considerations, and give the existence and regularity
theorems for the bounded viscosity solutions of the generalized Dirichlet
problems of second order fully nonlinear elliptic equations on the general
bounded domains, which may be irregular. The approximation method, the accretive
operator technique and the Caffarelli's perturbation theory are used.
Keywords:Pucci operator, viscosity solution, existence, $C^{2,\psi}$ regularity, Dini condition, fully nonlinear equation, general domain, accretive operator, approximation lemma Categories:35D05, 35D10, 35J60, 35J67 |
16. CJM 2002 (vol 54 pp. 634)
Frames and Single Wavelets for Unitary Groups We consider a unitary representation of a discrete countable abelian
group on a separable Hilbert space which is associated to a cyclic
generalized frame multiresolution analysis. We extend Robertson's
theorem to apply to frames generated by the action of the group.
Within this setup we use Stone's theorem and the theory of projection
valued measures to analyze wandering frame collections. This yields a
functional analytic method of constructing a wavelet from a
generalized frame multi\-resolution analysis in terms of the frame
scaling vectors. We then explicitly apply our results to the action
of the integers given by translations on $L^2({\mathbb R})$.
Keywords:wavelet, multiresolution analysis, unitary group representation, frame Categories:42C40, 43A25, 42C15, 46N99 |
17. CJM 2000 (vol 52 pp. 1149)
Canonical Resolution of a Quasi-ordinary Surface Singularity We describe the embedded resolution of an irreducible quasi-ordinary
surface singularity $(V,p)$ which results from applying the canonical
resolution of Bierstone-Milman to $(V,p)$. We show that this process
depends solely on the characteristic pairs of $(V,p)$, as predicted
by Lipman. We describe the process explicitly enough that a resolution
graph for $f$ could in principle be obtained by computer using only
the characteristic pairs.
Keywords:canonical resolution, quasi-ordinary singularity Categories:14B05, 14J17, 32S05, 32S25 |
18. CJM 2000 (vol 52 pp. 613)
Small Solutions of $\phi_1 x_1^2 + \cdots + \phi_n x_n^2 = 0$ Let $\phi_1,\dots,\phi_n$ $(n\geq 2)$ be nonzero integers such that
the equation
$$
\sum_{i=1}^n \phi_i x_i^2 = 0
$$
is solvable in integers $x_1,\dots,x_n$ not all zero. It is shown
that there exists a solution satisfying
$$
0 < \sum_{i=1}^n |\phi_i| x_i^2 \leq 2 |\phi_1 \cdots \phi_n|,
$$
and that the constant 2 is best possible.
Keywords:small solutions, diagonal quadratic forms Category:11E25 |
19. CJM 2000 (vol 52 pp. 123)
An Algorithm for Fat Points on $\mathbf{P}^2 Let $F$ be a divisor on the blow-up $X$ of $\pr^2$ at $r$ general
points $p_1, \dots, p_r$ and let $L$ be the total transform of a
line on $\pr^2$. An approach is presented for reducing the
computation of the dimension of the cokernel of the natural map
$\mu_F \colon \Gamma \bigl( \CO_X(F) \bigr) \otimes \Gamma \bigl(
\CO_X(L) \bigr) \to \Gamma \bigl( \CO_X(F) \otimes \CO_X(L) \bigr)$
to the case that $F$ is ample. As an application, a formula for
the dimension of the cokernel of $\mu_F$ is obtained when $r = 7$,
completely solving the problem of determining the modules in
minimal free resolutions of fat point subschemes\break
$m_1 p_1 + \cdots + m_7 p_7 \subset \pr^2$. All results hold for
an arbitrary algebraically closed ground field~$k$.
Keywords:Generators, syzygies, resolution, fat points, maximal rank, plane, Weyl group Categories:13P10, 14C99, 13D02, 13H15 |
20. CJM 1999 (vol 51 pp. 673)
Brownian Motion and Harmonic Analysis on Sierpinski Carpets We consider a class of fractal subsets of $\R^d$ formed in a manner
analogous to the construction of the Sierpinski carpet. We prove a
uniform Harnack inequality for positive harmonic functions; study
the heat equation, and obtain upper and lower bounds on the heat
kernel which are, up to constants, the best possible; construct a
locally isotropic diffusion $X$ and determine its basic properties;
and extend some classical Sobolev and Poincar\'e inequalities to
this setting.
Keywords:Sierpinski carpet, fractal, Hausdorff dimension, spectral dimension, Brownian motion, heat equation, harmonic functions, potentials, reflecting Brownian motion, coupling, Harnack inequality, transition densities, fundamental solutions Categories:60J60, 60B05, 60J35 |
21. CJM 1997 (vol 49 pp. 798)
Boundedness of solutions of parabolic equations with anisotropic growth conditions In this paper, we consider the parabolic equation
with anisotropic growth conditions, and obtain some criteria on
boundedness of solutions, which generalize the corresponding results
for the isotropic case.
Keywords:Parabolic equation, anisotropic growth conditions, generalized, solution, boundness Categories:35K57, 35K99. |