## Abstract

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 Poincar'e 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.

Original language | English |
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Pages (from-to) | 35-75 |

Number of pages | 41 |

Journal | Transactions of the American Mathematical Society |

Volume | 360 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2008 Jan |

## Keywords

- Asymptotic compactness
- CAT(0)-space
- G-convergence
- Gromov-Hausdorff convergence
- Harmonic map
- Lp-mapping space
- Measured metric space
- Mosco convergence
- Resolvent
- Spectrum
- The Poincar'e inequality

## ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics