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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:M rarr

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 rarr

N where t_n=2^-nt₀ and > 0. We prove that for any mapping f:M rarr

N the approximations E(f) are increasing in n isin

N provided the metric space (N,d) has curvature

-. 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 K

0) and the limit coincides with a lower semicontinuous functional on N (independent of ) 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

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