Monotonicity formulas in potential theory

  • Virginia Agostiniani
  • Lorenzo MazzieriEmail author


Using the electrostatic potential u due to a uniformly charged body \(\Omega \subset {\mathbb {R}}^n\), \(n\ge 3\), we introduce a family of monotone quantities associated with the level set flow of u. The derived monotonicity formulas are exploited to deduce a new quantitative version of the classical Willmore inequality.

Mathematics Subject Classification

35B06 53C21 35N25 



The author are grateful to G. Crasta, A. Farina, I. Fragalà, C. Mantegazza, J. Metzger, M. Novaga, and D. Peralta-Salas for useful comments and discussions during the preparation of the paper. The authors are members of Gruppo Nazionale per l’Analisi Matematica, la Probabilitàe le loro Applicazioni (GNAMPA), which is part of the Istituto Nazionale di Alta Matematica (INdAM), and partially funded by the GNAMPA project “Principi di fattorizzazione, formule di monotonia e disuguaglianze geometriche”. The authors would like to thank the anonymous referee for the careful reading of the manuscript as well as for his/her valuable suggestions.


  1. 1.
    Agostiniani, V., Fogagnolo, M., Mazzieri, L.: Minkowski inequalities via nonlinear potential theory. arXiv:1906.00322
  2. 2.
    Agostiniani, V., Fogagnolo, M., Mazzieri, L.: Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature. arXiv:1812.05022
  3. 3.
    Agostiniani, V., Mazzieri, L.: Riemannian aspects of potential theory. J. Math. Pures Appl. 104(3), 561–586 (2015)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Agostiniani, V., Mazzieri, L.: Comparing monotonicity formulas for electrostatic potentials and static metrics. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 28(1), 7–20 (2017)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Agostiniani, V., Mazzieri, L.: On the geometry of the level sets of bounded static potentials. Commun. Math. Phys. 355(1), 261–301 (2017)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Borghini, S., Mascellani, G., Mazzieri, L.: Some sphere theorems in linear potential theory. Trans. Am. Math. Soc. 371(11), 7757–7790 (2019)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Borghini, S., Mazzieri, L.: On the mass of static metrics with positive cosmological constant: II. arXiv:1711.07024
  8. 8.
    Borghini, S., Mazzieri, L.: On the mass of static metrics with positive cosmological constant: I. Class. Quantum Gravity 35(12), 125001 (2018)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Bour, V., Carron, G.: Optimal integral pinching results. Ann. Sci. Éc. Norm. Supér. (4) 48(1), 41–70 (2015)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Bray, H.L., Miao, P.: On the capacity of surfaces in manifolds with nonnegative scalar curvature. Invent. Math. 172(3), 459–475 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Brendle, S., Hung, P.-K., Wang, M.-T.: A Minkowski inequality for hypersurfaces in the Anti-de Sitter–Schwarzschild manifold. Commun. Pure Appl. Math. 69(1), 124–144 (2016)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Caffarelli, L.A., Friedman, A.: Partial regularity of the zero-set of solutions of linear and superlinear elliptic equations. J. Differ. Equ. 60(3), 420–433 (1985)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Chang, S.-Y.A., Wang, Y.: On Aleksandrov–Fenchel inequalities for k-convex domains. Milan J. Math. 79(1), 13 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Chang, S.-Y.A., Wang, Y.: Inequalities for quermassintegrals on k-convex domains. Adv. Math. 248, 335–377 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Cheeger, J., Naber, A., Valtorta, D.: Critical sets of elliptic equations. arXiv:1207.4236v3
  16. 16.
    Chen, B.-Y.: On a theorem of Fenchel–Borsuk–Willmore–Chern–Lashof. Math. Ann. 194(1), 19–26 (1971)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Chen, B.-Y.: On the total curvature of immersed manifolds, I: an inequality of Fenchel–Borsuk–Willmore. Am. J. Math. 93(1), 148–162 (1971)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Colding, T.H.: New monotonicity formulas for Ricci curvature and applications. I. Acta Math. 209(2), 229–263 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Colding, T.H., Minicozzi, W.P.: Monotonicity and its analytic and geometric implications. Proc. Natl. Acad. Sci. 110(48), 19233–19236 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    Colding, T.H., Minicozzi, W.P.: Ricci curvature and monotonicity for harmonic functions. Calc. Var. Partial. Differ. Equ. 49(3), 1045–1059 (2014)CrossRefMathSciNetzbMATHGoogle Scholar
  21. 21.
    Crasta, G., Fragalà, I., Gazzola, F.: On a long-standing conjecture by Pólya–Szegö and related topics. Z. Angew. Math. Phys. 56(5), 763–782 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  22. 22.
    DeTurck, D., Kazdan, J.L.: Some regularity theorems in Riemannian geometry. Ann. Scie. l’École Norm. Supérieure Ser. 4 14(3), 249–260 (1981)CrossRefMathSciNetzbMATHGoogle Scholar
  23. 23.
    Enciso, A., Peralta-Salas, D.: Symmetry for an overdetermined boundary problem in a punctured domain. Nonlinear Anal. 70(2), 1080–1086 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  24. 24.
    Farina, A., Mari, L., Valdinoci, E.: Splitting theorems, symmetry results and overdetermined problems for Riemannian manifolds. Commun. Partial Differ. Equ. 38(10), 1818–1862 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  25. 25.
    Federer, H.: Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften, vol. 153. Springer, New York (1969)Google Scholar
  26. 26.
    Fogagnolo, M., Mazzieri, L., Pinamonti, A.: Geometric aspects of p-capacitary potentials. Ann. Inst. H. Poincaré Anal. Non Linéaire 36(4), 1151–1179 (2019)CrossRefMathSciNetzbMATHGoogle Scholar
  27. 27.
    Fragalà, I., Gazzola, F.: Partially overdetermined elliptic boundary value problems. J. Differ. Equ. 245(5), 1299–1322 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  28. 28.
    Fragalà, I., Gazzola, F., Kawohl, B.: Overdetermined problems with possibly degenerate ellipticity, a geometric approach. Math. Z. 254(1), 117–132 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  29. 29.
    Freire, A., Schwartz, F.: Mass-capacity inequalities for conformally flat manifolds with boundary. Commun. Partial Differ. Equ. 39(1), 98–119 (2014)CrossRefMathSciNetzbMATHGoogle Scholar
  30. 30.
    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(2), 137–161 (1999)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Gerhardt, C.: Flow of nonconvex hypersurfaces into spheres. J. Differ. Geom. 32(1), 299–314 (1990)CrossRefMathSciNetzbMATHGoogle Scholar
  32. 32.
    Guan, P., Li, J.: The quermassintegral inequalities for k-convex starshaped domains. Adv. Math. 221(5), 1725–1732 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  33. 33.
    Hardt, R., Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T., Nadirashvili, N.: Critical sets of solutions to elliptic equations. J. Differ. Geom. 51(2), 359–373 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  34. 34.
    Hardt, R., Simon, L.: Nodal sets for solutions of elliptic equations. J. Differ. Geom. 30(2), 505–522 (1989)CrossRefMathSciNetzbMATHGoogle Scholar
  35. 35.
    Huisken, G.: An isoperimetric concept for the mass in general relativity. Video. Accessed Mar 2009
  36. 36.
    Huisken, G.: Asymptotic behavior for singularities of the mean curvature flow. J. Differ. Geom. 31(1), 285–299 (1990)CrossRefMathSciNetzbMATHGoogle Scholar
  37. 37.
    Huisken, G., Ilmanen, T.: The inverse mean curvature flow and the Riemannian Penrose inequality. J. Differ. Geom. 59(3), 353–437 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  38. 38.
    Hurtado, A., Palmer, V., Ritoré, M.: Comparison results for capacity. Indiana Univ. Math. J. 61(2), 539–555 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  39. 39.
    Kellogg, O.D.: Foundations of Potential Theory. Springer, Berlin (1967)CrossRefzbMATHGoogle Scholar
  40. 40.
    Marques, F.C., Neves, A.: Min–Max theory and the Willmore conjecture. Ann. Math. 179(2), 683–782 (2014)CrossRefMathSciNetzbMATHGoogle Scholar
  41. 41.
    Moser, R.: The inverse mean curvature flow and p-harmonic functions. J. Eur. Math. Soc. 9(1), 77–83 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  42. 42.
    Payne, L.E., Philippin, G.A.: On some maximum principles involving harmonic functions and their derivatives. SIAM J. Math. Anal. 10(1), 96–104 (1979)CrossRefMathSciNetzbMATHGoogle Scholar
  43. 43.
    Payne, L.E., Philippin, G.A.: Some overdetermined boundary value problems for harmonic functions. Z. Angew. Math. Phys. 42(6), 864–873 (1991)CrossRefMathSciNetzbMATHGoogle Scholar
  44. 44.
    Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math/0211159
  45. 45.
    Souc̆ek, J., Souc̆ek, V.: Morse–Sard theorem for real-analytic functions. Comment. Math. Univ. Carol. 13(1), 45–51 (1972)MathSciNetGoogle Scholar
  46. 46.
    Urbas, J.I.E.: On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures. Math. Z. 205(3), 355–372 (1990)CrossRefMathSciNetzbMATHGoogle Scholar
  47. 47.
    Willmore, T .J.: Mean curvature of immersed surfaces. Ann. Şti. Univ. “All. I. Cuza” Iaşi Secţ. I a Mat. (N.S.) 14, 99–103 (1968)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Xiao, J.: P-capacity vs surface-area. Adv. Math. 308, 1318–1336 (2017)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Università degli Studi di VeronaVeronaItaly
  2. 2.Università degli Studi di TrentoPovoItaly

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