Skip to main content
SpringerLink
Log in

Flux & radii within the subconformal capacity

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

This paper utilizes the p-capacity-flux and the p-capacity-radius to make an intrinsic three-fold exploration of the p-capacity for the subconformal case \(1<p<n\) in the Euclidean space \({\mathbb {R}}^n\). Even in the case \(p=2\) the here-established Theorems 1.2–2.2–3.2 are novel.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adams, D.R.: Choquet integrals in potential theory. Publ. Mat. 42, 3–66 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Adams, D.R., Hedberg, L.I.: Function Spaces and Potential Theory. Springer, Berlin (1996)

    Book  MATH  Google Scholar 

  3. Adams, D.R., Meyers, N.G.: Bessel potentials. Inclusion relations among classes of exceptional sets. Indiana Univ. Math. J. 22(1972/1973), 873–905

  4. Agostiniani, V., Fogagnolo, M., Mazzieri, L.: Minkowski inequalities via nonlinear potential theory. arXiv: 1906.00322v3 [math.AP] 3 Jan (2020)

  5. Agostiniani, V., Mazzieri, L.: Riemannian aspects of potential theory. J. Math. Pures Appl. 104, 561–586 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Agostiniani, V., Mazzieri, L.: On the geometry of the level sets of bounded static potentials. Commun. Math. Phys. 355, 261–301 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  7. Agostiniani, V., Mazzieri, L.: Monotonicity formulas in potential theory. Calc. Var. 59(1), Art. 6, 32 (2020)

  8. Akman, M., Gong, J., Hineman, J., Lewis, J., Vogel, A.: The Brunn-Minkowski inequality and a Minkowski problem for nonlinear capacity. Memoirs Am. Math. Soc. (to appear)

  9. Bianchini, C., Ciraolo, G.: A note on an overdetermined problem for the capacitary potential. Geometric Properties for Parabolic and Elliptic PDE’s, Vol. 176 (2016) of the series Springer Proc. Math. Stat. 41–48

  10. Bianchini, C., Ciraolo, G.: Wulff shape characterizations in overdetermined anisotropic elliptic problems. Comm. Partial Differ. Equ. (to appear)

  11. Bianchini, C., Ciraolo, G., Salani, P.: An overdetermined problem for the anisotropic capacity. Calc. Var. 55(84), 1–24 (2016)

    MathSciNet  MATH  Google Scholar 

  12. Bonnafé, A.: Estimates and asymptotic expansions for condenser \(p\)-capacities. The anisotropic case of segments. 2013\(\langle hal-00805229\rangle \)

  13. Böröczky, K., Lutwak, E., Yang, D., Zhang, G.: The logarithmic Minkowski problem. J. Am. Math. Soc. 26, 831–852 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bray, H.L., Miao, P.: On the capacity of surfaces in manifolds with nonnegative scalar curvature. Invent. Math. 172, 459–475 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bucur, D., Fragalá, I., Lamboley, J.: Optimal convex shapes for concave functionals. ESAIM: COCV 18, 693–711 (2012)

  16. Caffarelli, L.A., Jerison, D., Lieb, E.H.: On the case of equality in the Brunn-Minkowski inequality for capacity. Adv. Math. 117, 193–207 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ciska-Niedziałomska, M., Niedziałomska, K.: Stable foliations with respect to Fuglede modulus and level sets of \(p\)-harmonic functions. J. Math. Anal. Appl. 427, 440–459 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Colesanti, A., Nyström, K., Salani, P., Xiao, J., Yang, D., Zhang, G.: The Hadamard variational formula and the Minkowski problem for \(p\)-capacity. Adv. Math. 285, 1511–1588 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Colesanti, A., Salani, P.: The Brunn-Minkowski inequality for \(p\)-capacity of convex bodies. Math. Ann. 327, 459–479 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Crasta, G., Fragalá, I., Gazzola, F.: On a long-standing conjecture by Pólya-Szegö and related topics. Z. Angew. Math. Phys. 56, 763–782 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Dal Maso, G., Garroni, A.: Capacity theory for non-symmetric elliptic opereators. Ref. S.I.S.S.A. 179/M (Nov 1993)

  22. Della Pietra, F., Gavitone, N., Xia, C.: Motion of level sets by inverse anisotropic mean curvature. Commun. Anal. Geom. (to appear)

  23. Diaz, J.I.: Nonlinear Partial Differential Equations and Free Boundaries. Elliptic Equations, vol. I. Pitman Publishing, London (1985)

    MATH  Google Scholar 

  24. Druet, O.: Isoperimetric inequalities on nonpositively curved spaces. Preprint

  25. Evans, L.C.: The \(1\)-Laplacian, the \(\infty \)-Laplacian and differential games. Perspectives in nonlinear partial differential equations. Contemp. Math. 446(2007), 245–254

  26. Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions, Revised edition. CRC Press (2015)

  27. Federer, H., Ziemer, W.P.: The Lebesgue set of a function whose distribution derivatives are \(p\)th power summable. Indiana Univ. Math. J. 22, 139–158 (1973)

    Article  MATH  Google Scholar 

  28. Flucher, M.: Variational Problems with Concentration. Birkhäuser (1999)

  29. Fogagnolo, M., Mazzieri, L., Pinamonti, A.: Geometric aspects of \(p\)-capacity-potentials. Ann. Inst. H. Poincaré Anal. Non Linéaire 36, 1151–1179 (2019)

  30. Fragalá, I., Gazzola, F., Pierre, M.: On an isoperimetric inequality for capacity conjectured by Pólya and Szegö. J. Differ. Equ. 250, 1500–1520 (2011)

    Article  MATH  Google Scholar 

  31. Frehse, J.: Capacity methods in the theory of partial differential equations. Jber. d. Dt. Math.-Verein 84, 1–44 (1982)

  32. Freire, A., Schwartz, F.: Mass-capacity inequalities for conformally flat manifolds with boundary. Commun. Partial Differ. Equ. 39, 98–119 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  33. Gardner, R.J., Hartenstine, D.: Capacities, surface area, and radial sums. Adv. Math. 221, 601–626 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  34. Garofalo, N., Sartori, E.: Symmetry in exterior boundary value problems for quasilinear elliptic equations via blow-up and a priori estimates. Adv. Differ. Equ. 4, 137–161 (1999)

    MathSciNet  MATH  Google Scholar 

  35. Geroch, R.: Energy extraction. Ann. New York Acad. Sci. 224, 108–117 (1973)

    Article  MATH  Google Scholar 

  36. Grigor’yan, A.: Isoperimetric inequalities and capacities on Riemannian manifolds. The Maz’ya anniversary collection, Vol. 1 (Rostock, 1998), pp. 139–153, Oper. Theory Adv. Appl., 109, Birkhäuser, Basel (1999)

  37. Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degeneratio Elliptic Equations. Oxford University Press, Oxford (1993)

    MATH  Google Scholar 

  38. Hurtado, A., Palmer, V., Ritoré, M.: Comparison results for capacity. Indiana Univ. Math. J. 61, 539–555 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  39. Jauregui, J.L.: The capacity-volume inequality of Poincaré-Faber-Sezgö. Preprint

  40. Jerison, D.: Prescribing harmonic measure on convex domains. Invent. Math. 105, 375–400 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  41. Jerison, D.: A Minkowski problem for electrostatic capacity. Acta Math. 176, 1–47 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  42. Jerison, D.: The direct method in the calculus of variations for convex bodies. Adv. Math. 122, 262–279 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  43. Kawohl, B.: Variations on the \(p\)-Laplacian. In: Nonlinear Elliptic Partial Differential Equations. Contemporary Math. 540, 35–46 (2011)

  44. Korevaar, N.: Convexity of level sets for solutions to elliptic ring problems. Commun. Partial Differ. Equ. 15, 541–556 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  45. Ludwig, M., Xiao, J., Zhang, G.: Sharp convex Lorentz-Sobolev inequalities. Math. Ann. 350, 169–197 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  46. Lutwak, E.: A general Bieberbach inequality. Math. Proc. Camb. Phil. Soc. 78, 493–495 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  47. Lutwak, E.: The Brunn-Minkowski-Firey theorey I: mixed volumes and the Minkowski problem. J. Differ. Geom. 38, 131–150 (1993)

    Article  MATH  Google Scholar 

  48. Lutwak, E., Yang, D., Zhang, G.: On the \(L_p\)-Minkowski problem. Trans. Am. Math. Soc. 356, 4359–4370 (2003)

    Article  MATH  Google Scholar 

  49. Maz’ya, V.: Sobolev Spaces with Applications to Elliptic Partial Differential Equations, 2nd revised and augmented edition. Springer (2011)

  50. Mendez, O., Reichel, W.: Electrostatic characterization of spheres. Forum Math. 12, 223–245 (2000)

    MathSciNet  MATH  Google Scholar 

  51. Moser, R.: The inverse mean-curvature flow and \(p\)-harmonic functions. J. Eur. Math. Soc. 9, 77–83 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  52. Moser, R.: Geroch monotonicity and the construction of weak solutions of the inverse mean-curvature flow. Asian J. Math. 19, 357–376 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  53. Osserman, R.: Bonnesen-stype isoperimetric inequalities. Am. Math. Mon. 86, 1–29 (1979)

    Article  MATH  Google Scholar 

  54. Poggesi, G.: Radial symmetry for \(p\)-harmonic functions in exterior and punctured domains. Appl. Anal. 98, 1785–1798 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  55. Pólya, G.: Estimating electrostatic capacity. Am. Math. Mon. 54, 201–206 (1947)

    Article  MathSciNet  MATH  Google Scholar 

  56. Pólya, G., Szegö, G.: Inequalities for the capacity of a condenser. Am. J. Math. 67, 1–32 (1945)

    Article  MathSciNet  MATH  Google Scholar 

  57. Pólya, G., Szegö, G.: Isoperimetric Inequalities in Mathematical Physics. Princeton University Press, Princeton (1951)

    MATH  Google Scholar 

  58. Sangwine-Yager, J.R.: A Nonnesen-style inradius inequality in 3-space. Pac. J. Math. 134, 173–178 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  59. Sarvas, J.: Symmetrization of condensers in \(n\)-space. Ann. Acad. Sci. Fenn. Ser. AI 522, 1–44 (1972)

    MathSciNet  MATH  Google Scholar 

  60. Schneider, R.: Convex Bodies: The Brunn-Minkowski Theory. In: Encyclopedia of Math. and Its Appl. Vol 44. Cambridge University Press, Cambridge (1993)

  61. Schoen, R., Yau, S.-T.: Lectures on Differential Geometry. Conference Proceedings and Lecture Notes in Geometry and Topology, Vol. I, International Press (1994)

  62. Shubin, M.: Capacity and its applications. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.140.1404&rep=rep1&type=pdf

  63. Tee, G.J.: Surface area and capacity of ellipsoids in n dimensions. New Zealand J. Math. 34, 165–198 (2005)

    MathSciNet  MATH  Google Scholar 

  64. Troyanov, M.: Parabolicity of manifolds. Siberian Adv. Math. 9, 125–150 (1999)

    MathSciNet  MATH  Google Scholar 

  65. Tso, K.S.: A direct method approach for the existence of convex hypersurfaces with prescribed Gauss-Kronecker curvature. Math. Z. 209, 339–334 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  66. Xiao, J.: The \(p\)-harmonic capacity of an asymptotically flat \(3\)-manifold with nonnegative scalar curvature. Ann. H. Pincaré 17, 2265–2283 (2016)

    Article  MATH  Google Scholar 

  67. Xiao, J.: Entropy flux—electrostatic capacity—graphical mass. Proc. Am. Math. Soc. 145, 825–832 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  68. Xiao, J.: A maximum problem of S.-T. Yau for variational \(p\)-capacity. Adv. Geom. 17, 483–496 (2017)

  69. Xiao, J.: P-capacity versus surface-area. Adv. Math. 308, 1318–1336 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  70. Xiao, J.: The \(p\)-affine capacity redux. J. Geom. Anal. 27, 2872–2888 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  71. Xiao, J.: Prescribing capacitary curvature measures on planar convex domains. J. Geom. Anal. 30, 2225–2240 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  72. Xu, X.: Some results on functional capacity and their applications to \(p\)-Laplacian problems involving measure data. Nonlinear Anal. 27, 17–36 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  73. Yau, S.-T.: Problem section. In: S.-T. Yau (ed.) Seminar on Differential Geometry, pp. 669-706. Ann. Math. Stud. 102, Princeton University Press, Princeton, NJ (1982)

Download references

Acknowledgements

The authors wish to thank the managing editor A. Malchiodi and the anonymous referee for their suggestions about how to present this paper.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Memorial University, St. John’s, NL, A1C 5S7, Canada

    Jie Xiao

  2. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, Hubei, China

    Ning Zhang

Authors
  1. Jie Xiao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ning Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jie Xiao.

Additional information

Communicated by A. Malchiodi.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Jie Xiao is supported by NSERC of Canada (#20171864); Ning Zhang is supported by NSF of China (# 11901217 and # 11971005).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xiao, J., Zhang, N. Flux & radii within the subconformal capacity. Calc. Var. 60, 120 (2021). https://doi.org/10.1007/s00526-021-01989-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-021-01989-5

Mathematics Subject Classification

Search

Navigation