Variational convergence over metric spaces
Variational convergence over metric spaces
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DOI：
10.1090/s0002-9947-07-04167-0
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发表时间：
2005-05
影响因子：
1.3
通讯作者：
K. Kuwae;T. Shioya
中科院分区：
文献类型：
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作者：
K. Kuwae;T. Shioya
在定义域是度量空间关于度量Gromov-Hausdorff拓扑的收敛序列且目标是Gromov-Hausdorff收敛序列的情况下，给出了映射Lp-收敛的一个自然定义，p> 1.在Lp-收敛的基础上，建立了变分收敛理论.证明了Poincare不等式在附加条件下蕴含渐近紧性。渐近紧性等价于能量子水平集的Gromov-Hausdorff紧性。假设目标是CAT（0）-空间，我们研究了预解式的收敛性。作为应用，我们研究了度量空间上的能量泛函的逼近以及能量泛函在Ricci曲率下界下的收敛性。
We introduce a natural definition of L p -convergence of maps, p > 1, in the case where the domain is a convergent sequence of measured metric space with respect to the measured Gromov-Hausdorff topology and the target is a Gromov-Hausdorff convergent sequence. With the L p -convergence, we establish a theory of variational convergences. We prove that the Poincare inequality with some additional condition implies the asymptotic compactness. The asymptotic compactness is equivalent to the Gromov-Hausdorff compactness of the energy-sublevel sets. Supposing that the targets are CAT(0)-spaces, we study convergence of resolvents. As applications, we investigate the approximating energy functional over a measured metric space and convergence of energy functionals with a lower bound of Ricci curvature.