Variational convergence over metric spaces

Kazuhiro Kuwae, Takashi Shioya

Research output: Contribution to journalArticlepeer-review

10 Citations (Scopus)


We introduce a natural definition of Lp-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 Lp-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 languageEnglish
Pages (from-to)35-75
Number of pages41
JournalTransactions of the American Mathematical Society
Issue number1
Publication statusPublished - 2008 Jan


  • 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


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