首页 | 本学科首页   官方微博 | 高级检索  
     检索      


Monotone Approximation of Energy Functionals for Mappings into Metric Spaces -- II
Authors:Sturm  KT
Abstract:We investigate approximations E(f) of energy functionals E(f) for generalized harmonic maps f:MrarrN between singular spaces. Given any symmetric submarkovian semigroup (P) on any measure space (M, 
$$M$$
,m) and any metric space (N,d) we study the approximated energy functionals

$$E^{n,0} (f) = \frac{1}{{2t_n }}\smallint _M \smallint _M d^2 (f(x),f(y))P_{t_n } (x,dy)m(dx),$$
as well as

$$E^{n,\kappa } (f) = \frac{1}{{\kappa t_n }}\smallint _M \smallint _M \log \cosh (\sqrt \kappa \cdot d(f(x),f(y)))P_{t_n } (x,dy)m(dx).$$
for mappings f:MrarrN where tn=2-nt0 and kappa > 0. We prove that for any mapping f:MrarrN the approximations E(f) are increasing in nisinN provided the metric space (N,d) has curvature ge-kappa. Moreover, for any symmetric submarkovian semigroup (P) which is associated with a strongly local, quasi-regular Dirichlet form and for any bounded L2-mapping f:M rarr N the approximations E(f) converge (for all Kge0) and the limit coincides with a lower semicontinuous functional on N (independent of kappa) provided the metric space (N,d) has relatively compact balls and {lower bounded curvature}.
Keywords:Dirichlet form  harmonic map  energy form  singular space  Alexandrov space  
本文献已被 SpringerLink 等数据库收录!
设为首页 | 免责声明 | 关于勤云 | 加入收藏

Copyright©北京勤云科技发展有限公司  京ICP备09084417号