Potential Analysis

, Volume 11, Issue 4, pp 359–386

Monotone Approximation of Energy Functionals for Mappings into Metric Spaces -- II

  • K.T. Sturm

DOI: 10.1023/A:1008664108526

Cite this article as:
Sturm, K. Potential Analysis (1999) 11: 359. doi:10.1023/A:1008664108526


We investigate approximations E(f) of energy functionals E(f) for generalized harmonic maps f:M→N 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:M→N where tn=2-nt0 and κ > 0. We prove that for any mapping f:M→N the approximations E(f) are increasing in n∈N provided the metric space (N,d) has curvature ≥-κ. 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 → N the approximations E(f) converge (for all K≥0) and the limit coincides with a lower semicontinuous functional on N (independent of κ) provided the metric space (N,d) has relatively compact balls and {lower bounded curvature}.
Dirichlet form harmonic map energy form singular space Alexandrov space. 

Copyright information

© Kluwer Academic Publishers 1999

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  • K.T. Sturm

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