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Isovolumetric and Isoperimetric Problems for a Class of Capillarity Functionals

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Abstract

Capillarity functionals are parameter invariant functionals defined on classes of two-dimensional parametric surfaces in \({\mathbb{R}^{3}}\) as the sum of the area integral and an anisotropic term of suitable form. In the class of parametric surfaces with the topological type of \({\mathbb{S}^{2}}\) and with fixed volume, extremals of capillarity functionals are surfaces whose mean curvature is prescribed up to a constant. For a certain class of anisotropies vanishing at infinity, we prove the existence and nonexistence of volume-constrained, \({\mathbb{S}^{2}}\)-type, minimal surfaces for the corresponding capillarity functionals. Moreover, in some cases, we show the existence of extremals for the full isoperimetric inequality.

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References

  1. Aubin, Th.: Nonlinear Analysis on Manifolds. Monge-Ampère Equations, Grundlehren der Mathematischen Wissenschaften, vol. 252. Springer, New York, 1982

  2. Bangert V.: The existence of gaps in minimal foliations. Aequationes Math. 34, 153–166 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bethuel F., Ghidaglia J.M.: Improved regularity of solutions to elliptic equations involving Jacobians and applications. J. Math. Pures Appl. 72, 441–474 (1993)

    MATH  MathSciNet  Google Scholar 

  4. Bononcini V.: Un teorema di continuità per integrali su superficie chiuse. Rivista Mat. Univ. Parma 4, 299–311 (1953)

    MATH  MathSciNet  Google Scholar 

  5. Brezis H., Coron J.M.: Convergence of solutions of H-systems or how to blow bubbles. Arch. Rat. Mech. Anal. 89, 21–56 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  6. Caffarelli L., de la Llave R.: Planelike Minimizers in Periodic Media. Commun. Pure Appl. Math. 54, 1403–1441 (2001)

    Article  MATH  Google Scholar 

  7. Caldiroli P.: H-bubbles with prescribed large mean curvature. Manuscripta Math. 113, 125–142 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  8. Caldiroli P.: Blow-up analysis for the prescribed mean curvature equation on \({\mathbb{R}^{2}}\). J. Funct. Anal. 257, 405–427 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  9. Caldiroli P., Musina R.: Existence of minimal H-bubbles. Commun. Contemp. Math. 4, 177–209 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  10. Caldiroli P., Musina R.: H-bubbles in a perturbative setting: the finite-dimensional reduction method. Duke Math. J. 122, 457–484 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  11. Caldiroli P., Musina R.: The Dirichlet problem for H-systems with small boundary data: blowup phenomena and nonexistence results. Arch. Ration. Mech. Anal. 181, 1–42 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. Caldiroli, P., Musina, R.: Weak limit and blowup of approximate solutions to H-systems. J. Funct. Anal. 249, 171–198 (2007)

  13. Caldiroli, P., Musina, R.: Bubbles with prescribed mean curvature: the variational approach. Nonlinear Anal. Theory Methods Appl. Ser. A 74, 2985–2999 (2011)

  14. Dierkes, U., Hildebrandt, S., Sauvigny, F.: Minimal Surfaces, GMW 339. Springer 2010

  15. Ekeland, I.: Nonconvex minimization problems. Bull. Am. Math. Soc. 1, 443–474 (1979)

  16. Felli, V.: A note on the existence of H-bubbles via perturbation methods. Rev. Mat. Iberoamer. 21, 163–178 (2005)

  17. Goldman, M., Novaga, M.: Volume-constrained minimizers for the prescribed curvature problem in periodic media. Calc. Var. PDE 44, 297–318 (2012)

  18. Gulliver, R.D., Osserman, R., Royden, H.L.: A theory of branched immersions of surfaces. Am. J. Math. 95, 750–812 (1973)

  19. Heinz, E.: Über die Regularität schwacher Lösungen nichtlinearer elliptisher Systeme. Nachr. Akad. Wiss. Gottingen II. Math. Phys. Kl. 1, 1–15 (1986)

  20. Hildebrandt, S., von der Mosel, H.: Conformal representation of surfaces, and Plateau’s problem for Cartan functionals. Riv. Mat. Univ. Parma (7) 4 *, 1–43 (2005)

  21. Musina, R.: The role of the spectrum of the Laplace operator on \({\mathbb{S}^2}\) in the H-bubble problem. J. Anal. Math. 94, 265-291 (2004)

  22. Steffen K.: Isoperimetric inequalities and the problem of Plateau. Math. Ann. 222, 97–144 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  23. Struwe, M.: Plateau’s problem and the calculus of variations. Princeton University Press, 2014 (originally published in 1989)

  24. Topping P.: The optimal constant in Wente’s L ∞ estimate. Comment. Math. Helv. 72, 316–328 (1977)

    Article  MathSciNet  Google Scholar 

  25. Wente H.: An existence theorem for surfaces of constant mean curvature. J. Math. Anal. Appl. 26, 318–344 (1969)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Torino, via Carlo Alberto, 10 - 10123, Torino, Italy

    Paolo Caldiroli

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  1. Paolo Caldiroli
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Correspondence to Paolo Caldiroli.

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Communicated by P. Rabinowitz

Work partially supported by the PRIN-2012-74FYK7 Grant “Variational and perturbative aspects of nonlinear differential problems”, by the project ERC Advanced Grant 2013 n. 339958 “Complex Patterns for Strongly Interacting Dynamical Systems—COMPAT”, and by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

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Caldiroli, P. Isovolumetric and Isoperimetric Problems for a Class of Capillarity Functionals. Arch Rational Mech Anal 218, 1331–1361 (2015). https://doi.org/10.1007/s00205-015-0881-y

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