Variational convergence over metric spaces

Kuwae, Kazuhiro; Shioya, Takashi

doi:10.1090/S0002-9947-07-04167-0

Variational convergence over metric spaces
HTML articles powered by AMS MathViewer

by Kazuhiro Kuwae and Takashi Shioya PDF

Trans. Amer. Math. Soc. 360 (2008), 35-75 Request permission

Abstract:

We introduce a natural definition of $L^p$-convergence of maps, $p \ge 1$, 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 Poincaré 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 $\operatorname {CAT}(0)$-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.

References

S. G. Bobkov, Isoperimetric and analytic inequalities for log-concave probability measures, Ann. Probab. 27 (1999), no. 4, 1903–1921. MR 1742893, DOI 10.1214/aop/1022874820
Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486, DOI 10.1007/978-3-662-12494-9
Peter Buser, A note on the isoperimetric constant, Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 2, 213–230. MR 683635
J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999), no. 3, 428–517. MR 1708448, DOI 10.1007/s000390050094
Jeff Cheeger and Tobias H. Colding, On the structure of spaces with Ricci curvature bounded below. III, J. Differential Geom. 54 (2000), no. 1, 37–74. MR 1815411
Gianni Dal Maso, An introduction to $\Gamma$-convergence, Progress in Nonlinear Differential Equations and their Applications, vol. 8, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1201152, DOI 10.1007/978-1-4612-0327-8
Kenji Fukaya, Collapsing of Riemannian manifolds and eigenvalues of Laplace operator, Invent. Math. 87 (1987), no. 3, 517–547. MR 874035, DOI 10.1007/BF01389241
Misha Gromov, Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics, vol. 152, Birkhäuser Boston, Inc., Boston, MA, 1999. Based on the 1981 French original [ MR0682063 (85e:53051)]; With appendices by M. Katz, P. Pansu and S. Semmes; Translated from the French by Sean Michael Bates. MR 1699320
Piotr Hajłasz and Pekka Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), no. 688, x+101. MR 1683160, DOI 10.1090/memo/0688
Jürgen Jost, Equilibrium maps between metric spaces, Calc. Var. Partial Differential Equations 2 (1994), no. 2, 173–204. MR 1385525, DOI 10.1007/BF01191341
Jürgen Jost, Convex functionals and generalized harmonic maps into spaces of nonpositive curvature, Comment. Math. Helv. 70 (1995), no. 4, 659–673. MR 1360608, DOI 10.1007/BF02566027
Jürgen Jost, Generalized harmonic maps between metric spaces, Geometric analysis and the calculus of variations, Int. Press, Cambridge, MA, 1996, pp. 143–174. MR 1449406, DOI 10.1016/0167-5273(96)02633-2
Jürgen Jost, Nonpositive curvature: geometric and analytic aspects, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1997. MR 1451625, DOI 10.1007/978-3-0348-8918-6
—, Nonlinear Dirichlet forms, New directions in Dirichlet forms, Amer. Math. Soc., Providence, RI, 1998, pp. 1–47.
A. Kasue, Convergence of Riemannian manifolds and harmonic maps, preprint, 2000.
Atsushi Kasue, Convergence of Riemannian manifolds and Laplace operators. II, Potential Anal. 24 (2006), no. 2, 137–194. MR 2217418, DOI 10.1007/s11118-005-8568-x
—, Convergence of metric graphs and energy forms, preprint, 2004.
Alexander V. Kolesnikov, Mosco convergence of Dirichlet forms in infinite dimensions with changing reference measures, J. Funct. Anal. 230 (2006), no. 2, 382–418. MR 2186217, DOI 10.1016/j.jfa.2005.06.002
Alexander V. Kolesnikov, Convergence of Dirichlet forms with changing speed measures on $\Bbb R^d$, Forum Math. 17 (2005), no. 2, 225–259. MR 2130907, DOI 10.1515/form.2005.17.2.225
Nicholas J. Korevaar and Richard M. Schoen, Sobolev spaces and harmonic maps for metric space targets, Comm. Anal. Geom. 1 (1993), no. 3-4, 561–659. MR 1266480, DOI 10.4310/CAG.1993.v1.n4.a4
Pekka Koskela, Nageswari Shanmugalingam, and Jeremy T. Tyson, Dirichlet forms, Poincaré inequalities, and the Sobolev spaces of Korevaar and Schoen, Potential Anal. 21 (2004), no. 3, 241–262. MR 2075670, DOI 10.1023/B:POTA.0000033331.88514.6e
Kazuhiro Kuwae and Takashi Shioya, Convergence of spectral structures: a functional analytic theory and its applications to spectral geometry, Comm. Anal. Geom. 11 (2003), no. 4, 599–673. MR 2015170, DOI 10.4310/CAG.2003.v11.n4.a1
Kazuhiro Kuwae and Takashi Shioya, Sobolev and Dirichlet spaces over maps between metric spaces, J. Reine Angew. Math. 555 (2003), 39–75. MR 1956594, DOI 10.1515/crll.2003.014
Zhi Ming Ma and Michael Röckner, Introduction to the theory of (nonsymmetric) Dirichlet forms, Universitext, Springer-Verlag, Berlin, 1992. MR 1214375, DOI 10.1007/978-3-642-77739-4
Uwe F. Mayer, Gradient flows on nonpositively curved metric spaces and harmonic maps, Comm. Anal. Geom. 6 (1998), no. 2, 199–253. MR 1651416, DOI 10.4310/CAG.1998.v6.n2.a1
R. E. Megginson, An introduction to Banach space theory, Graduate Texts in Mathematics, vol. 183, Springer-Verlag, New York, 1998.
Umberto Mosco, Composite media and asymptotic Dirichlet forms, J. Funct. Anal. 123 (1994), no. 2, 368–421. MR 1283033, DOI 10.1006/jfan.1994.1093
Y. Ogura, T. Shioya, and M. Tomisaki, Collapsing of warped product spaces and one-dimensional diffusion processes, in preparation.
Shin-ichi Ohta, Cheeger type Sobolev spaces for metric space targets, Potential Anal. 20 (2004), no. 2, 149–175. MR 2032946, DOI 10.1023/A:1026359313080
K. T. Sturm, Diffusion processes and heat kernels on metric spaces, Ann. Probab. 26 (1998), no. 1, 1–55. MR 1617040, DOI 10.1214/aop/1022855410
Constantin Vernicos, The macroscopic sound of tori, Pacific J. Math. 213 (2004), no. 1, 121–156. MR 2040254, DOI 10.2140/pjm.2004.213.121