Abstract

We study a problem of minimal surfaces with free boundary written in the form of a non convex minimization problem. Our aim is to characterize optimal solutions by finding a suitable calibration field. A natural upper bound of the infimum is given by a variant of the Cheeger problem that we solve explicitly proving the optimality thanks to the construction of a cut-locus potential. The comparison with the original problem is then discussed in detail.

Author information

Contact details are reproduced from the original publication and may be historical.

Minh Phan

The University of Danang, University of Science and Education, Danang, Vietnam

ptdminh@ued.udn.vn

Keywords

  • Free boundary problems
  • calibrations
  • minimal surfaces
  • shape derivative
  • Cheeger sets
  • cut-locus potential

Mathematics Subject Classification

49J45, 49N15, 49Q10, 65K10, 90C26

References

  1. F. Alter, V. Caselles: Uniqueness of the Cheeger set of a convex body, Nonlinear Analysis 70 (2009) 32–44.https://doi.org/10.1016/j.na.2007.11.032Google Scholar
  2. F. Alter, V. Caselles, A. Chambolle: A characterization of convex calibrable sets in RN, Math. Annalen 332 (2005) 329–366.https://doi.org/10.1007/s00208-004-0628-9Google Scholar
  3. L. Ambrosio, N. Fusco, D. Pallara: Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, Oxford (2000).https://doi.org/10.1093/oso/9780198502456.001.0001Google Scholar
  4. G. Anzellotti: Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl. (4) 135 (1983) 293–318.https://doi.org/10.1007/BF01781073Google Scholar
  5. G. Anzellotti: Traces of bounded vector fields and the divergence theorem, preprint, Univ. of Trento.Google Scholar
  6. G. Anzellotti, M. Giaquinta: Funzioni BV e tracce, Rend. Sem. Mat. Univ. Padova 60 (1978) 1–21.Google Scholar
  7. H. Attouch, G. M. Buttazzo, G. Michaille: Variational Analysis in Sobolev and BV Spaces, MPS/SIAM Series on Optimization Vol. 6, Society of Industrial and Applied Mathematics, Philadelphia (2006).Google Scholar
  8. H. Attouch, T. Champion: Lp regularization of the non-parametric minimal surface problem, in: Ill-Posed Variational Problems and Regularization Techniques, Trier 1998, M. Théra et al. (eds.), Lecture Notes in Economics and Mathematical Systems 477, Springer, Berlin (1999) 25–34.https://doi.org/10.1007/978-3-642-45780-7_2Google Scholar
  9. G. Bouchitté, I. Fragalà: Duality theory for non-convex problems in the calculus of variations, Arch. Ration. Mech. Anal. 229 (2018) 361–415.https://doi.org/10.1007/s00205-018-1219-3Google Scholar
  10. G. Bouchitté, I. Fragala, I. Lucardesi: A variational method for second order shape derivatives, SIAM J. Control Optim. 54/2 (2016) 1056–1084.https://doi.org/10.1137/15100494XGoogle Scholar
  11. G. Bouchitté, M. Phan: A convex relaxation method for free boundary problems, Evol. Equations Control Theory 17 (2026) 62–92.https://doi.org/10.3934/eect.2025015Google Scholar
  12. H. Brezis: Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer Science & Business Media, Cham (2010).https://doi.org/10.1007/978-0-387-70914-7Google Scholar
  13. G. Buttazzo, G. Carlier, M. Comte: On the selection of maximal Cheeger sets, Diff. Int. Equations 20 (2007) 991–1004.https://doi.org/10.57262/die/1356039307Google Scholar
  14. V. Caselles, A. Chambolle, M. Novaga: Uniqueness of the Cheeger set of a convex body, Pacific J. Math. 232 (2007) 77–90.https://doi.org/10.2140/pjm.2007.232.77Google Scholar
  15. G. Crasta, I. Fragalà: On the Dirichlet and Serrin problems for the inhomogeneous infinity Laplacian in convex domains: Regularity and geometric results, Arch. Ration. Mech. Anal. 218 (2015) 1577–1607.https://doi.org/10.1007/s00205-015-0888-4Google Scholar
  16. G. Crasta, I. Fragalà: Characterization of stadium-like domains via boundary value problems for the infinity Laplacian, Nonlinear Analysis 133 (2016) 228–249.https://doi.org/10.1016/j.na.2015.12.007Google Scholar
  17. G. Crasta, I. Fragalà: On the characterization of some classes of proximally smooth sets, ESAIM Control, Optim. Calc. Var. 22 (2016) 710–727.https://doi.org/10.1051/cocv/2015022Google Scholar
  18. G. Crasta, I. Fragalà: Geometric issues in PDE problems related to the infinity Laplace operator, in: Topological Optimization and Optimal Transport in the Applied Sciences, M. Bergounioux et al. (eds.), Radon Series on Computational and Applied Mathematics Vol. 17, De Gruyter, Berlin (2017) 5–19.https://doi.org/10.1515/9783110430417-001Google Scholar
  19. I. Ekeland, R. Temam: Analyse Convexe et Problèmes Variationnels, Dunod Gauthier-Villars, Paris (1974).Google Scholar
  20. L. C. Evans, R. F. Gariepy: Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton (1992).Google Scholar
  21. A. Gray: Tubes, Progress in Mathematics Vol. 221, Birkhäuser, Basel (2012).Google Scholar
  22. A. Henrot, M. Pierre: Variation et Optimisation de Formes. Une Analyse Géométrique, Matématiques et Applications 48, Springer, Berlin (2005).https://doi.org/10.1007/3-540-37689-5Google Scholar
  23. B. Kawohl, T. Lachand-Robert: Characterization of Cheeger sets for convex subsets of the plane, Pacific J. Math. 225 (2006) 103–118.https://doi.org/10.2140/pjm.2006.225.103Google Scholar
  24. G. P. Leonardi: An overview on the Cheeger problem, in: New Trends in Shape Optimization, Springer, Berlin (2015) 117–139.https://doi.org/10.1007/978-3-319-17563-8_6Google Scholar
  25. Y. Li, L. Nirenberg: The distance function to the boundary, Finsler geometry, and the singular set of viscosity solutions of some Hamilton-Jacobi equations, Comm. Pure Appl. Math. 58 (2005) 85–146.https://doi.org/10.1002/cpa.20051Google Scholar
  26. E. Parini: An introduction to the Cheeger roblem, Surveys Math. Appl. 6 (2011) 9–21.Google Scholar
  27. R. Neumayer, G. Saracco: The Cheeger constant of a Jordan domain without necks, Calc. Var. Partial Diff. Equations 56/6 (2017), art.no. 164, 29p.https://doi.org/10.1007/s00526-017-1263-0Google Scholar
  28. R. Neumayer, G. Saracco: Minimizers of the prescribed curvature functional in a Jordan domain with no necks, ESAIM Control Optim. Calc. Var. 26 (2020), art.no. 76, 20p.https://doi.org/10.1051/cocv/2020030Google Scholar
  29. P.-M. Suquet: Fonctions à variation bornée sur un ouvert de RN, Travaux Sém. Analyse Convexe 8/1 (1978), exp.no. 4, 17 pp.Google Scholar
  30. E. Stredulinsky, W. P. Ziemer: Area minimizing sets subject to a volume constraint in a convex set, J. Geom. Analysis 7 (1997) 653–677.https://doi.org/10.1007/BF02921639Google Scholar

Suggested citation

G. Bouchitté, M. Phan. “Calibrations for Minimal Surfaces with Free Boundary and Cheeger-Type Problems.” Journal of Convex Analysis 33 (2026), No. 3&4, 691–736. https://doi.org/10.68381/jca33041

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Published by Heldermann Verlag, 2026. Rights now held by Banach Press LLC.