Abstract
In this paper, we consider a large class of Bernoulli-type free boundary problems with mixed periodic-Dirichlet boundary conditions. We show that solutions with non-flat profile can be found variationally as global minimizers of the classical Alt–Caffarelli energy functional.
Funding source: Grantová Agentura, Univerzita Karlova
Award Identifier / Grant number: PRIMUS/19/SCI/01
Award Identifier / Grant number: UNCE/SCI/023
Funding source: Office of International Science and Engineering
Award Identifier / Grant number: 0967140
Funding source: Division of Mathematical Sciences
Award Identifier / Grant number: 1412095
Award Identifier / Grant number: 1714098
Funding statement: The authors acknowledge the Center for Nonlinear Analysis (NSF PIRE Grant no. OISE-0967140) where part of this work was carried out. The research of the authors was partially funded by the National Science Foundation under Grant nos. DMS-1412095 and DMS-1714098. The first author also acknowledges the support of the research support programs of Charles University under Grant nos. PRIMUS/19/SCI/01 and UNCE/SCI/023.
Acknowledgements
This paper is part of the first author’s Ph.D. thesis at Carnegie Mellon University. The second author would like to thank Ovidiu Savin and Eugen Varvaruca for their helpful insights. The authors would also like to thank Luis Caffarelli, Ming Chen, Craig Evans and Ian Tice for useful conversations on the subject of this paper.
References
[1] H. W. Alt and L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. 325 (1981), 105–144. 10.1515/crll.1981.325.105Search in Google Scholar
[2] H. W. Alt, L. A. Caffarelli and A. Friedman, Jet flows with gravity, J. Reine Angew. Math. 331 (1982), 58–103. 10.1515/crll.1982.331.58Search in Google Scholar
[3] H. W. Alt, L. A. Caffarelli and A. Friedman, Axially symmetric jet flows, Arch. Ration. Mech. Anal. 81 (1983), no. 2, 97–149. 10.1007/BF00250648Search in Google Scholar
[4] H. W. Alt, L. A. Caffarelli and A. Friedman, A free boundary problem for quasilinear elliptic equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 11 (1984), no. 1, 1–44. Search in Google Scholar
[5] H. W. Alt, L. A. Caffarelli and A. Friedman, Jets with two fluids. I. One free boundary, Indiana Univ. Math. J. 33 (1984), no. 2, 213–247. 10.1512/iumj.1984.33.33011Search in Google Scholar
[6] H. W. Alt, L. A. Caffarelli and A. Friedman, Variational problems with two phases and their free boundaries, Trans. Amer. Math. Soc. 282 (1984), no. 2, 431–461. 10.1090/S0002-9947-1984-0732100-6Search in Google Scholar
[7] H. W. Alt, L. A. Caffarelli and A. Friedman, Abrupt and smooth separation of free boundaries in flow problems, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 12 (1985), no. 1, 137–172. Search in Google Scholar
[8] C. J. Amick and L. E. Fraenkel, On the behavior near the crest of waves of extreme form, Trans. Amer. Math. Soc. 299 (1987), no. 1, 273–298. 10.1090/S0002-9947-1987-0869412-4Search in Google Scholar
[9] C. J. Amick, L. E. Fraenkel and J. F. Toland, On the Stokes conjecture for the wave of extreme form, Acta Math. 148 (1982), 193–214. 10.1007/BF02392728Search in Google Scholar
[10] D. Arama and G. Leoni, On a variational approach for water waves, Comm. Partial Differential Equations 37 (2012), no. 5, 833–874. 10.1080/03605302.2012.661819Search in Google Scholar
[11]
L. A. Caffarelli,
A Harnack inequality approach to the regularity of free boundaries. I. Lipschitz free boundaries are
[12] L. A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. III. Existence theory, compactness, and dependence on X, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 15 (1988), no. 4, 583–602. Search in Google Scholar
[13] L. A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. II. Flat free boundaries are Lipschitz, Comm. Pure Appl. Math. 42 (1989), no. 1, 55–78. 10.1002/cpa.3160420105Search in Google Scholar
[14] L. A. Caffarelli, D. Jerison and C. E. Kenig, Global energy minimizers for free boundary problems and full regularity in three dimensions, Noncompact Problems at the Intersection of Geometry, Analysis, and Topology, Contemp. Math. 350, American Mathematical Society, Providence (2004), 83–97. 10.1090/conm/350/06339Search in Google Scholar
[15] H. Chang-Lara and O. Savin, Boundary regularity for the free boundary in the one-phase problem, New Developments in the Analysis of Nonlocal Operators, Contemp. Math. 723, American Mathematical Society, Providence (2019), 149–165. 10.1090/conm/723/14549Search in Google Scholar
[16] R. M. Chen, S. Walsh and M. H. Wheeler, On the existence and qualitative theory of stratified solitary water waves, C. R. Math. Acad. Sci. Paris 354 (2016), no. 6, 601–605. 10.1016/j.crma.2016.03.004Search in Google Scholar
[17] R. M. Chen, S. Walsh and M. H. Wheeler, Existence and qualitative theory for stratified solitary water waves, Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 2, 517–576. 10.1016/j.crma.2016.03.004Search in Google Scholar
[18] A. Constantin, D. Sattinger and W. Strauss, Variational formulations for steady water waves with vorticity, J. Fluid Mech. 548 (2006), 151–163. 10.1017/S0022112005007469Search in Google Scholar
[19] A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math. 57 (2004), no. 4, 481–527. 10.1002/cpa.3046Search in Google Scholar
[20] A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math. 63 (2010), no. 4, 533–557. 10.1002/cpa.20299Search in Google Scholar
[21] A. Constantin, W. Strauss and E. Vărvărucă, Global bifurcation of steady gravity water waves with critical layers, Acta Math. 217 (2016), no. 2, 195–262. 10.1007/s11511-017-0144-xSearch in Google Scholar
[22] D. Danielli and A. Petrosyan, A minimum problem with free boundary for a degenerate quasilinear operator, Calc. Var. Partial Differential Equations 23 (2005), no. 1, 97–124. 10.1007/s00526-004-0294-5Search in Google Scholar
[23] D. De Silva and D. Jerison, A singular energy minimizing free boundary, J. Reine Angew. Math. 635 (2009), 1–21. 10.1515/CRELLE.2009.074Search in Google Scholar
[24] N. Edelen and M. Engelstein, Quantitative stratification for some free-boundary problems, Trans. Amer. Math. Soc. 371 (2019), no. 3, 2043–2072. 10.1090/tran/7401Search in Google Scholar
[25] I. Fonseca, G. Leoni and M. G. Mora, A second order minimality condition for a free-boundary problem, Ann. Sc. Norm. Super. Pisa Cl. Sci., to appear. 10.2422/2036-2145.201706_014Search in Google Scholar
[26] L. E. Fraenkel, A constructive existence proof for the extreme Stokes wave, Arch. Ration. Mech. Anal. 183 (2007), no. 2, 187–214. 10.1007/s00205-006-0003-ySearch in Google Scholar
[27] A. Friedman, Variational principles and free-boundary problems, 2nd ed., Robert E. Krieger, Malabar, 1988. Search in Google Scholar
[28] G. Gravina, Variational techniques for water waves and singular perturbations, Ph.D Thesis, Carnegie Mellon University, 2019. Search in Google Scholar
[29] G. Gravina and G. Leoni, On the behavior of the free boundary for a one-phase Bernoulli problem with mixed boundary conditions, preprint (2019), https://arxiv.org/abs/1910.14643. 10.3934/cpaa.2020215Search in Google Scholar
[30] D. Jerison and O. Savin, Some remarks on stability of cones for the one-phase free boundary problem, Geom. Funct. Anal. 25 (2015), no. 4, 1240–1257. 10.1007/s00039-015-0335-6Search in Google Scholar
[31] B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Math. 1150, Springer, Berlin, 1985. 10.1007/BFb0075060Search in Google Scholar
[32] G. Keady and J. Norbury, On the existence theory for irrotational water waves, Math. Proc. Cambridge Philos. Soc. 83 (1978), no. 1, 137–157. 10.1017/S0305004100054372Search in Google Scholar
[33] R. H. Kinsey and S. Wu, A priori estimates for two-dimensional water waves with angled crests, Camb. J. Math. 6 (2018), no. 2, 93–181. 10.4310/CJM.2018.v6.n2.a1Search in Google Scholar
[34] J. P. Krasovskiĭ, On the theory of steady-state waves of finite amplitude, Ž. Vyčisl. Mat i Mat. Fiz. 1 (1961), 836–855. 10.1016/0041-5553(62)90025-3Search in Google Scholar
[35] J. B. McLeod, The asymptotic behavior near the crest of waves of extreme form, Trans. Amer. Math. Soc. 299 (1987), no. 1, 299–302. 10.1090/S0002-9947-1987-0869413-6Search in Google Scholar
[36] J. B. McLeod, The Stokes and Krasovskii conjectures for the wave of greatest height, Stud. Appl. Math. 98 (1997), no. 4, 311–333. 10.1111/1467-9590.00051Search in Google Scholar
[37] L. M. Milne-Thomson, Theoretical Hydrodynamics, 4th ed., The Macmillan, New York, 1960. Search in Google Scholar
[38] P. I. Plotnikov, Proof of the Stokes conjecture in the theory of surface waves, Stud. Appl. Math. 108 (2002), no. 2, 217–244. 10.1111/1467-9590.01408Search in Google Scholar
[39] P. I. Plotnikov and J. F. Toland, Convexity of Stokes waves of extreme form, Arch. Ration. Mech. Anal. 171 (2004), no. 3, 349–416. 10.1007/s00205-003-0292-3Search in Google Scholar
[40] G. G. Stokes, Considerations relative to the greatest height of oscillatory irrotational waves which can be propagated without change of form, Math. Phys. Paper 1 (1880), 225–228. Search in Google Scholar
[41] J. F. Toland, On the existence of a wave of greatest height and Stokes’s conjecture, Proc. Roy. Soc. London Ser. A 363 (1978), no. 1715, 469–485. 10.1098/rspa.1978.0178Search in Google Scholar
[42] J. F. Toland, Non-existence of global energy minimisers in Stokes waves problems, Discrete Contin. Dyn. Syst. 34 (2014), no. 8, 3211–3217. 10.3934/dcds.2014.34.3211Search in Google Scholar
[43] E. Varvaruca and G. S. Weiss, A geometric approach to generalized Stokes conjectures, Acta Math. 206 (2011), no. 2, 363–403. 10.1007/s11511-011-0066-ySearch in Google Scholar
[44] G. S. Weiss, Partial regularity for a minimum problem with free boundary, J. Geom. Anal. 9 (1999), no. 2, 317–326. 10.1007/BF02921941Search in Google Scholar
[45] G. S. Weiss, Boundary monotonicity formulae and applications to free boundary problems. I. The elliptic case, Electron. J. Differential Equations 2004 (2004), Paper No. 44. Search in Google Scholar
[46] G. S. Weiss and G. Zhang, A free boundary approach to two-dimensional steady capillary gravity water waves, Arch. Ration. Mech. Anal. 203 (2012), no. 3, 747–768. 10.1007/s00205-011-0466-3Search in Google Scholar
© 2022 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- In memoriam Emmanuele DiBenedetto (1947–2021)
- Local regularity results for solutions of linear elliptic equations with drift term
- On the existence of non-flat profiles for a Bernoulli free boundary problem
- On the blow-up of GSBV functions under suitable geometric properties of the jump set
- Anisotropic liquid drop models
- Rigidity and trace properties of divergence-measure vector fields
Articles in the same Issue
- Frontmatter
- In memoriam Emmanuele DiBenedetto (1947–2021)
- Local regularity results for solutions of linear elliptic equations with drift term
- On the existence of non-flat profiles for a Bernoulli free boundary problem
- On the blow-up of GSBV functions under suitable geometric properties of the jump set
- Anisotropic liquid drop models
- Rigidity and trace properties of divergence-measure vector fields