Eigenfunctions of the Laplace-Beltrami Operator, and Isoperimetric and Isocapacitary Inequalities

Andrea Cianchi

Bollettino dell'Unione Matematica Italiana (2013)

  • Volume: 6, Issue: 1, page 167-190
  • ISSN: 0392-4041

How to cite

top

Cianchi, Andrea. "Eigenfunctions of the Laplace-Beltrami Operator, and Isoperimetric and Isocapacitary Inequalities." Bollettino dell'Unione Matematica Italiana 6.1 (2013): 167-190. <http://eudml.org/doc/294030>.

@article{Cianchi2013,
author = {Cianchi, Andrea},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {167-190},
publisher = {Unione Matematica Italiana},
title = {Eigenfunctions of the Laplace-Beltrami Operator, and Isoperimetric and Isocapacitary Inequalities},
url = {http://eudml.org/doc/294030},
volume = {6},
year = {2013},
}

TY - JOUR
AU - Cianchi, Andrea
TI - Eigenfunctions of the Laplace-Beltrami Operator, and Isoperimetric and Isocapacitary Inequalities
JO - Bollettino dell'Unione Matematica Italiana
DA - 2013/2//
PB - Unione Matematica Italiana
VL - 6
IS - 1
SP - 167
EP - 190
LA - eng
UR - http://eudml.org/doc/294030
ER -

References

top
  1. ALVINO, A. - CIANCHI, A. - MAZ'YA, V. G. - MERCALDO, A., Well-posed elliptic Neumann problems involving irregular data and domains, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 1017-1054 Zbl1200.35105MR2659156DOI10.1016/j.anihpc.2010.01.010
  2. BAIDER, A., Noncompact Riemannian manifolds with discrete spectra, J. Diff. Geom.14 (1979), 41-57. Zbl0411.58022MR577877
  3. BERGER, M. - GAUDUCHON, P. - MAZET, E., Le spectre d'une variété Riemannienne, Lecture notes in Mathematics194, Springer-Verlag, Berlin, 1971. MR282313
  4. BENJAMINI, I. - CAO, J., A new isoperimetric theorem for surfaces of variable curvature, Duke Math. J.85 (1996), 359-396. Zbl0886.53031MR1417620DOI10.1215/S0012-7094-96-08515-4
  5. BIRMAN, M. S. - SOLOMJAK, M. Z., Spectral theory of self-adjoint operators in Hilbert space, D. Reidel Publishing Company, Dordrecht, 1986. MR1192782
  6. BOURGAIN, J., Geodesic restrictions and L p -estimates for eigenfunctions of Riemannian surfaces, Amer. Math. Soc. Tranl.226 (2009), 27-25. Zbl1189.58015MR2500507DOI10.1090/trans2/226/03
  7. BROOKS, B., On the spectrum of non-compact manifolds with finite volume, Math. Zeit.9 (1984), 425-432. Zbl0537.58040MR757481DOI10.1007/BF01161957
  8. BROOKS, B., The bottom of the spectrum of a Riemannian covering, J. Reine Angew. Math.357 (1985), 101-114. Zbl0553.53027MR783536DOI10.1515/crll.1985.357.101
  9. BURAGO, YU. D. - ZALGALLER, V. A., Geometric inequalities, Springer-Verlag, Berlin, 1988. MR936419DOI10.1007/978-3-662-07441-1
  10. BURENKOV, V. I. - DAVIES, E. B., Spectral stability of the Neumann Laplacian, J. Diff. Eq.186 (2002), 485-508. Zbl1042.35035MR1942219DOI10.1016/S0022-0396(02)00033-5
  11. CHAVEL, I., Eigenvalues in Riemannian geometry, Academic Press, New York, 1984. Zbl0551.53001MR768584
  12. CHAVEL, I. - FELDMAN, E. A., Modified isoperimetric constants, and large time heat diffusion in Riemannian manifolds, Duke Math. J.64 (1991), 473-499. Zbl0753.58031MR1141283DOI10.1215/S0012-7094-91-06425-2
  13. CHEEGER, J., A lower bound for the smallest eigevalue of the Laplacian, in Problems in analysis, 195-199, Princeton Univ. Press, Princeton, 1970. MR402831
  14. CHITI, G., A reverse Hölder inequality for the eigenfunctions of linear second order elliptic operators, Zeit. Angew. Math. Phys. (ZAMP) 33 (1982), 143-148. Zbl0508.35063MR652928DOI10.1007/BF00948319
  15. CHUNG, F. - GRIGOR'YAN, A. - YAU, S.-T., Higher eigenvalues and isoperimetric inequalities on Riemannian manifolds and graphs, Comm. Anal. Geom.8 (2000), 969-1026. Zbl1001.58022MR1846124DOI10.4310/CAG.2000.v8.n5.a2
  16. CIANCHI, A., On relative isoperimetric inequalities in the plane, Boll. Un. Mat. Ital.3-B (1989), 289-326. Zbl0674.49030MR997998
  17. CIANCHI, A., A sharp form of Poincaré type inequalities on balls and spheres, Z. Angew. Math. Phys. (ZAMP) 40 (1989), 558-569. Zbl0707.53034MR1008923DOI10.1007/BF00944807
  18. CIANCHI, A., Moser-Trudinger inequalities without boundary conditions and isoperimetric problems, Indiana Univ. Math. J.54 (2005), 669-705. Zbl1097.46016MR2151230DOI10.1512/iumj.2005.54.2589
  19. CIANCHI, A. - MAZ'YA, V. G., Neumann problems and isocapacitary inequalities, J. Math. Pures Appl.89 (2008), 71-105. Zbl1146.35041MR2378090DOI10.1016/j.matpur.2007.10.001
  20. CIANCHI, A. - MAZ'YA, V. G., On the discreteness of the spectrum of the Laplacian on complete Riemannian manifolds, J. Diff. Geom.87 (2011), 469-491. MR2819545
  21. CIANCHI, A. - MAZ'YA, V. G., Bounds for eigenfunctions of the Laplacian on noncompact Riemannian manifolds, Amer. J. Math., to appear. MR3068397DOI10.1353/ajm.2013.0028
  22. CIANCHI, A. - MAZ'YA, V. G., Boundedness of solutions to the Schrödinger equation under Neumann boundary conditions, J. Math. Pures Appl.98 (2012), 654-688. MR2994697DOI10.1016/j.matpur.2012.05.007
  23. COULHON, T. - GRIGOR'YAN, A. - LEVIN, D., On isoperimetric profiles of product spaces, Comm. Anal. Geom.11 (2003), 85-120. Zbl1085.53027MR2016197DOI10.4310/CAG.2003.v11.n1.a5
  24. COURANT, R. - HILBERT, D., Methoden der mathematischen Physik, Springer, Berlin, 1937. Zbl63.0449.05MR344038
  25. DAVIES, E. B. - SIMON, B., Spectral properties of the Neumann Laplacian of horns, Geom. Funct. Anal.2 (1992), 105-117. Zbl0749.35024MR1143665DOI10.1007/BF01895707
  26. DE GIORGI, E., Sulla proprietà isoperimetrica dell'ipersfera, nella classe degli insiemi aventi frontiera orientata di misura finita, (Italian) Atti Accad. Naz. Lincei. Mem. Cl. Sci. Fis. Mat. Nat. Sez. I 5 (1958) 33-44. Zbl0116.07901MR98331
  27. DONNELLY, H., Bounds for eigenfunctions of the Laplacian on compact Riemannian manifolds, J. Funct. Anal.187 (2001), 247-261. Zbl0991.58006MR1867351DOI10.1006/jfan.2001.3817
  28. DONNELLY, H., Eigenvalue estimates for certain noncompact manifolds, Michigan Math. J.31 (1984), 349-357. Zbl0591.58033MR767614DOI10.1307/mmj/1029003079
  29. DONNELLY, H. - LI, P., Pure point spectrum and negative curvature for noncompact manifolds, Duke Math. J.46 (1979), 497-503. Zbl0416.58025MR544241
  30. ESCOBAR, J. F., On the spectrum of the Laplacian on complete Riemannian manifolds, Comm. Part. Diff. Equat.11 (1986), 63-85. Zbl0585.58046MR814547DOI10.1080/03605308608820418
  31. GALLOT, S., Inégalités isopérimétriques et analitiques sur les variétés riemanniennes, Asterisque163 (1988), 31-91. MR999971
  32. GRIGOR'YAN, A., On the existence of positive fundamental solutions of the Laplace equation on Riemannian manifolds, Mat. Sbornik128 (1985), 354-363 (Russian); English translation: Math. USSR Sb.56 (1987), 349-358. 
  33. GRIGOR'YAN, A., Isoperimetric inequalities and capacities on Riemannian manifolds, in The Maz'ya anniversary collection, Vol. 1 (Rostock, 1998), 139-153, Oper. Theory Adv. Appl., 109, Birkhuser, Basel, 1999. MR1747869
  34. GRIMALDI, R. - PANSU, P., Calibrations and isoperimetric profiles, Amer. J. Math.129 (2007), 315-350. Zbl1121.53034MR2306037DOI10.1353/ajm.2007.0010
  35. HAIASZ, P. - KOSKELA, P., Isoperimetric inequalites and imbedding theorems in irregular domains, J. London Math. Soc.58 (1998), 425-450. MR1668136DOI10.1112/S0024610798006346
  36. HEBEY, E., Nonlinear analysis on manifolds: Sobolev spaces and inequalities, American Math. Soc., Providence, 1999. Zbl0981.58006MR1688256
  37. HEMPEL, R. - SECO, L. - SIMON, B., The essential spectrum of Neumann Laplacians on some bounded singular domains, J. Funct. Anal.102 (1991), 448-483. Zbl0741.35043MR1140635DOI10.1016/0022-1236(91)90130-W
  38. HOFFMANN-OSTENHOF, M. - HOFFMANN-OSTENHOF, T. - NADIRASHVILI, N., On the multiplicity of eigenvalues of the Laplacian on surfaces, Ann. Global Anal. Geom.17 (1999), 43-48. Zbl0923.35109MR1674331DOI10.1023/A:1006595115793
  39. JAKSIC, V. - MOLCHANOV, S. - SIMON, B., Eigenvalue asymptotics of the Neumann Laplacian of regions and manifolds with cusps, J. Funct. Anal.106 (1992), 59-79. Zbl0783.35040MR1163464DOI10.1016/0022-1236(92)90063-O
  40. KILPELÄINEN, T. - MALÝ, J., Sobolev inequalities on sets with irregular boundaries, Z. Anal. Anwendungen19 (2000), 369-380. MR1768998DOI10.4171/ZAA/956
  41. KLEINE, R., Discreteness conditions for the Laplacian on complete non-compact Riemannian manifolds, Math. Zeit.198 (1988), 127-141. Zbl0622.53024MR938034DOI10.1007/BF01183044
  42. KLEINE, R., Warped products with discrete spectra, Results Math.15 (1989), 81-103. Zbl0669.58030MR979446DOI10.1007/BF03322449
  43. KLEINER, B., An isoperimetric comparison theorem, Invent. Math.108 (1992), 37-47. Zbl0770.53031MR1156385DOI10.1007/BF02100598
  44. LABUTIN, D. A., Embedding of Sobolev spaces on Hölder domains, Proc. Steklov Inst. Math.227 (1999), 163-172 (Russian); English translation: Trudy Mat. Inst.227 (1999), 170-179. MR1784315
  45. LIONS, P.-L. - PACELLA, F., Isoperimetric inequalities for convex cones, Proc. Amer. Math. Soc.109 (1990), 477-485. Zbl0717.52008MR1000160DOI10.2307/2048011
  46. MALYÂ, J. - ZIEMER, W. P., Fine regularity of solutions of elliptic partial differential equations, American Mathematical Society, Providence, 1997. MR1461542DOI10.1090/surv/051
  47. MAZ'YA, V. G., Classes of regions and imbedding theorems for function spaces, Dokl. Akad. Nauk. SSSR133 (1960), 527-530 (Russian); English translation: Soviet Math. Dokl.1 (1960), 882-885. MR126152
  48. MAZ'YA, V. G., Some estimates of solutions of second-order elliptic equations, Dokl. Akad. Nauk. SSSR137 (1961), 1057-1059 (Russian); English translation: Soviet Math. Dokl.2 (1961), 413-415. MR131054
  49. MAZ'YA, V. G., On p-conductivity and theorems on embedding certain functional spaces into a C-space, Dokl. Akad. Nauk SSSR, 140 (1961), 299-302 (Russian). MR157224
  50. MAZ'YA, V. G., On the solvability of the Neumann problem, Dokl. Akad. Nauk SSSR, 147 (1962), 294-296 (Russian). MR144058
  51. MAZ'YA, V. G., The Neumann problem in regions with nonregular boundaries, Sibirsk. Mat. Ž.9 (1968), 1322-1350 (Russian). MR239270
  52. MAZ'YA, V. G., On weak solutions of the Dirichlet and Neumann problems, Trusdy Moskov. Mat. Obšč.20 (1969), 137-172 (Russian); English translation: Trans. Moscow Math. Soc.20 (1969), 135-172 MR259329
  53. MAZ'YA, V. G., Sobolev spaces with applications to elliptic partial differential equations, Springer, Heidelberg, 2011. MR2777530DOI10.1007/978-3-642-15564-2
  54. MORGAN, F. - HOWARDS, H. - HUTCHINGS, M., The isoperimetric problem on surfaces of revolution of decreasing Gauss curvature, Trans. Amer. Math. Soc, 352 (2000), 4889-4909. Zbl0976.53082MR1661278DOI10.1090/S0002-9947-00-02482-X
  55. MORGAN, F. - JOHNSON, D. L., Some sharp isoperimetric theorems for Riemannian manifolds, Indiana Univ. Math. J.49 (2000), 1017-1041. Zbl1021.53020MR1803220DOI10.1512/iumj.2000.49.1929
  56. NADIRASHVILI, N., Isoperimetric inequality for the second eigenvalue of a sphere, J. Diff. Geom.61 (2002), 335-340. Zbl1071.58024MR1972149
  57. PAYNE, L. E. - RAYNER, M. E., An isoperimetric inequality for the first eigenfunction in the fixed membrane problem, Z. Angew. Math. Phys.23 (1972), 13-15. Zbl0241.73080MR313649DOI10.1007/BF01593198
  58. PITTET, CH., The isoperimetric profile of homogeneous Riemannian manifolds, J. Diff. Geom.54 (2000), 255-302. Zbl1035.53069MR1818180
  59. REED, M. - SIMON, B., Analysis of Operators IV, Academic Press, New York, 1972. 
  60. RITORÉ, M. , Constant geodesic curvature curves and isoperimetric domains in rotationally symmetric surfaces, Comm. Anal. Geom.9 (2001), 1093-1138. Zbl1018.53003MR1883725DOI10.4310/CAG.2001.v9.n5.a5
  61. RÖLKE, W., Uber den Laplace-Operator auf Riemannschen Mannigfaltigkeiten mit diskontinuierlichen Gruppen, Math. Nachr.21 (1960), 132-149. 
  62. SALOFF-COSTE, L., Sobolev inequalities in familiar and unfamiliar settings, in Sobolev spaces in mathematics, Vol I, Sobolev type inequalities, V.G. Maz'ya editor, Springer, 2009. Zbl1165.26011MR2508847DOI10.1007/978-0-387-85648-3_11
  63. SMITH, H. F. - SOGGE, C. D., On the L p norm of spectral clusters for compact manifolds with boundary, Acta Math.198 (2007), 107-153. Zbl1189.58017MR2316270DOI10.1007/s11511-007-0014-z
  64. SOGGE, C. D., Lectures on eigenfunctions of the Laplacian, Topics in mathematical analysis, 337-360, Ser. Anal. Appl. Comput., 3, World Sci. Publ., Hackensack, NJ, 2008. Zbl1170.35071MR2462960DOI10.1142/9789812811066_0011
  65. SOGGE, C. D. - ZELDITCH, S., Riemannian manifolds with maximal eigenfunction growth, Duke Math. J.114 (2002), 387-437. Zbl1018.58010MR1924569DOI10.1215/S0012-7094-02-11431-8
  66. STRICHARTZ, R. S., Analysis of the Laplacian on complete Riemannian manifolds, J. Funct. Anal.52 (1983), 48-79. Zbl0515.58037MR705991DOI10.1016/0022-1236(83)90090-3
  67. YAU, S. T., Isoperimetric constants and the first eigenvalue of a compact manifold, Ann. Sci. Ecole Norm. Sup.8 (1975), 487-507. Zbl0325.53039MR397619
  68. ZIEMER, W. P., Weakly differentiable functions, Springer-Verlag, New York, 1989. Zbl0692.46022MR1014685DOI10.1007/978-1-4612-1015-3

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.