Expand all Collapse all | Results 1 - 6 of 6 |
1. CMB 2012 (vol 57 pp. 12)
On the Continuity of the Eigenvalues of a Sublaplacian We study the behavior of the eigenvalues of a sublaplacian $\Delta_b$ on a compact strictly pseudoconvex CR manifold $M$, as functions on the set
${\mathcal P}_+$ of positively oriented contact forms on $M$ by endowing ${\mathcal P}_+$ with a natural metric topology.
Keywords:CR manifold, contact form, sublaplacian, Fefferman metric Categories:32V20, 53C56 |
2. CMB 2011 (vol 56 pp. 3)
Semiclassical Limits of Eigenfunctions on Flat $n$-Dimensional Tori We provide a proof of a conjecture by Jakobson, Nadirashvili, and
Toth stating
that on an $n$-dimensional flat torus $\mathbb T^{n}$, and the Fourier transform
of squares of the eigenfunctions $|\varphi_\lambda|^2$ of the Laplacian have
uniform $l^n$ bounds that do not depend on the eigenvalue $\lambda$. The proof
is a generalization of an argument by Jakobson, et al. for the
lower dimensional cases. These results imply uniform bounds for semiclassical
limits on $\mathbb T^{n+2}$. We also prove a geometric lemma that bounds the number of
codimension-one simplices satisfying a certain restriction on an
$n$-dimensional sphere $S^n(\lambda)$ of radius $\sqrt{\lambda}$, and we use it in
the proof.
Keywords:semiclassical limits, eigenfunctions of Laplacian on a torus, quantum limits Categories:58G25, 81Q50, 35P20, 42B05 |
3. CMB 2008 (vol 51 pp. 217)
A Multivalued Nonlinear System with the Vector $p$-Laplacian on the Semi-Infinity Interval We study a second order nonlinear system driven by the vector
$p$-Laplacian, with a multivalued nonlinearity and defined on
the positive time semi-axis $\mathbb{R}_+.$ Using degree
theoretic techniques we solve an auxiliary mixed boundary value
problem defined on the finite interval $[0,n]$ and then via a
diagonalization method we produce a solution for the original
infinite time-horizon system.
Keywords:semi-infinity interval, vector $p$-Laplacian, multivalued nonlinear, fixed point index, Hartman condition, completely continuous map Category:34A60 |
4. CMB 2007 (vol 50 pp. 356)
Existence of Positive Solutions for Nonlinear Noncoercive Hemivariational Inequalities In this paper we investigate the existence of positive solutions
for nonlinear elliptic problems driven by the $p$-Laplacian with a
nonsmooth potential (hemivariational inequality). Under asymptotic
conditions that make the Euler functional indefinite and
incorporate in our framework the asymptotically linear problems,
using a variational approach based on nonsmooth critical point
theory, we obtain positive smooth solutions. Our analysis also
leads naturally to multiplicity results.
Keywords:$p$-Laplacian, locally Lipschitz potential, nonsmooth critical point theory, principal eigenvalue, positive solutions, nonsmooth Mountain Pass Theorem Categories:35J20, 35J60, 35J85 |
5. CMB 2006 (vol 49 pp. 358)
On the Principal Eigencurve of the $p$-Laplacian: Stability Phenomena We show that each point of the principal eigencurve of the
nonlinear problem
$$
-\Delta_{p}u-\lambda m(x)|u|^{p-2}u=\mu|u|^{p-2}u \quad
\text{in } \Omega,
$$
is stable (continuous) with respect to the exponent $p$ varying in
$(1,\infty)$; we also prove some convergence results
of the principal eigenfunctions corresponding.
Keywords:$p$-Laplacian with indefinite weight, principal eigencurve, principal eigenvalue, principal eigenfunction, stability Categories:35P30, 35P60, 35J70 |
6. CMB 2000 (vol 43 pp. 51)
Eigenfunction Decay For the Neumann Laplacian on Horn-Like Domains The growth properties at infinity for eigenfunctions corresponding to
embedded eigenvalues of the Neumann Laplacian on horn-like domains
are studied. For domains that pinch at polynomial rate, it is shown
that the eigenfunctions vanish at infinity faster than the reciprocal
of any polynomial. For a class of domains that pinch at an exponential
rate, weaker, $L^2$ bounds are proven. A corollary is that eigenvalues
can accumulate only at zero or infinity.
Keywords:Neumann Laplacian, horn-like domain, spectrum Categories:35P25, 58G25 |