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Search: All articles in the CJM digital archive with keyword energy

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1. CJM 2014 (vol 66 pp. 1110)

Li, Dong; Xu, Guixiang; Zhang, Xiaoyi
On the Dispersive Estimate for the Dirichlet Schrödinger Propagator and Applications to Energy Critical NLS
We consider the obstacle problem for the Schrödinger 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 Schrödinger 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 Schrödinger propagator, dispersive estimate, Dirichlet boundary condition, scattering theory, energy critical
Categories:35P25, 35Q55, 47J35

2. CJM 2013 (vol 66 pp. 1413)

Zhang, Xi; Zhang, Xiangwen
Generalized Kähler--Einstein Metrics and Energy Functionals
In this paper, we consider a generalized Kähler-Einstein equation on Kähler manifold $M$. Using the twisted $\mathcal K$-energy introduced by Song and Tian, we show that the existence of generalized Kähler-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 Kähler-Einstein metric implies a Moser-Trudinger type inequality.

Keywords:complex Monge--Ampère equation, energy functional, generalized Kähler--Einstein metric, Moser--Trudinger type inequality
Categories:53C55, 32W20

3. CJM 2011 (vol 64 pp. 24)

Borodachov, S. V.
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)

Teplyaev, Alexander
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)

Martínez-Finkelshtein, A.; Maymeskul, V.; Rakhmanov, E. A.; Saff, E. B.
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

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