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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 Online first
Generalized KÃ¤hler--Einstein Metrics and Energy Functionals In this paper, we consider a generalized
KÃ¤hler-Einstein equation on KÃ¤hler manifold $M$. Using the
twisted $\mathcal K$-energy introduced by Song and Tian, we show
that the existence of generalized KÃ¤hler-Einstein metrics with
semi-positive twisting $(1, 1)$-form $\theta $ is also closely
related to the properness of the twisted $\mathcal K$-energy
functional. Under the condition that the twisting form $\theta $ is
strictly positive at a point or $M$ admits no nontrivial Hamiltonian
holomorphic vector field, we prove that the existence of generalized
KÃ¤hler-Einstein metric implies a Moser-Trudinger type inequality.
Keywords:complex Monge--AmpÃ¨re equation, energy functional, generalized KÃ¤hler--Einstein metric, Moser--Trudinger type inequality Categories:53C55, 32W20 |
3. CJM 2011 (vol 64 pp. 24)
Lower Order Terms of the Discrete Minimal Riesz Energy on Smooth Closed Curves We consider the problem of minimizing the energy of $N$ points
repelling each other on curves in $\mathbb{R}^d$ with the potential
$|x-y|^{-s}$, $s\geq 1$, where $|\, \cdot\, |$ is
the Euclidean norm. For a sufficiently smooth, simple, closed,
regular curve, we find the next order term in the asymptotics of the
minimal $s$-energy. On our way, we also prove that at
least for $s\geq 2$, the minimal pairwise distance in optimal configurations
asymptotically equals $L/N$, $N\to\infty$, where $L$ is the length
of the curve.
Keywords:minimal discrete Riesz energy, lower order term, power law potential, separation radius Categories:31C20, 65D17 |
4. CJM 2008 (vol 60 pp. 457)
Harmonic Coordinates on Fractals with Finitely Ramified Cell Structure We define sets with finitely ramified cell structure, which are
generalizations of post-crit8cally finite self-similar
sets introduced by Kigami and of fractafolds introduced by Strichartz. In general,
we do not assume even local self-similarity, and allow countably many cells
connected at each junction point.
In particular, we consider post-critically infinite fractals.
We prove that if Kigami's resistance form
satisfies certain assumptions, then there exists a weak Riemannian metric
such that the energy can be expressed as the integral of the norm squared
of a weak gradient with respect to an energy measure.
Furthermore, we prove that if such a set can be homeomorphically represented
in harmonic coordinates, then for smooth functions the weak gradient can be
replaced by the usual gradient.
We also prove a simple formula for the energy measure Laplacian in harmonic
coordinates.
Keywords:fractals, self-similarity, energy, resistance, Dirichlet forms, diffusions, quantum graphs, generalized Riemannian metric Categories:28A80, 31C25, 53B99, 58J65, 60J60, 60G18 |
5. CJM 2004 (vol 56 pp. 529)
Asymptotics for Minimal Discrete Riesz Energy on Curves in $\R^d$ We consider the $s$-energy
$$
E(\ZZ_n;s)=\sum_{i \neq j} K(\|z_{i,n}-z_{j,n}\|;s)
$$
for point sets $\ZZ_n=\{ z_{k,n}:k=0,\dots,n\}$ on certain compact sets
$\Ga$ in $\R^d$ having finite one-dimensional Hausdorff measure, where
$$
K(t;s)=
\begin{cases}
t^{-s} ,& \mbox{if } s>0, \\
-\ln t, & \mbox{if } s=0,
\end{cases}
$$
is the Riesz kernel. Asymptotics for the minimum $s$-energy and the
distribution of minimizing sequences of points is studied. In
particular, we prove that, for $s\geq 1$, the minimizing nodes for a
rectifiable Jordan curve $\Ga$ distribute asymptotically uniformly with
respect to arclength as $n\to\infty$.
Keywords:Riesz energy, Minimal discrete energy,, Rectifiable curves, Best-packing on curves Categories:52A40, 31C20 |