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

Abstract

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 t_n=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}.