October  2020, 19(10): 4853-4878. doi: 10.3934/cpaa.2020215

On the behavior of the free boundary for a one-phase Bernoulli problem with mixed boundary conditions

1. 

Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic

2. 

Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, USA

* Corresponding author

Received  November 2019 Revised  June 2020 Published  July 2020

Fund Project: 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 G. Gravina and G. Leoni was partially funded by the National Science Foundation under Grants No. DMS-1412095 and DMS- 1714098. G. Gravina also acknowledges the support of the research support programs of Charles University: PRIMUS/19/SCI/01 and UNCE/SCI/023

This paper is concerned with the study of the behavior of the free boundary for a class of solutions to a two-dimensional one-phase Bernoulli free boundary problem with mixed periodic-Dirichlet boundary conditions. It is shown that if the free boundary of a symmetric local minimizer approaches the point where the two different conditions meet, then it must do so at an angle of $ \pi/2 $.

Citation: Giovanni Gravina, Giovanni Leoni. On the behavior of the free boundary for a one-phase Bernoulli problem with mixed boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4853-4878. doi: 10.3934/cpaa.2020215
References:
[1]

H. W. Alt, Strömungen durch inhomogene poröse Medien mit freiem Rand, J. Reine Angew. Math., 305 (1979), 89-115.  doi: 10.1515/crll.1979.305.89.  Google Scholar

[2]

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.   Google Scholar

[3]

H. W. AltL. A. Caffarelli and A. Friedman, Axially symmetric jet flows, Arch. Rational Mech. Anal., 81 (1983), 97-149.  doi: 10.1007/BF00250648.  Google Scholar

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H. W. AltL. A. Caffarelli and A. Friedman, Jets with two fluids. I. One free boundary, Indiana Univ. Math. J., 33 (1984), 213-247.  doi: 10.1512/iumj.1984.33.33011.  Google Scholar

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C. J. Amick and L. E. Fraenkel, On the behavior near the crest of waves of extreme form, Trans. Amer. Math. Soc., 299 (1987), 273-298.  doi: 10.2307/2000494.  Google Scholar

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C. J. AmickL. E. Fraenkel and J. F. Toland, On the Stokes conjecture for the wave of extreme form, Acta Math., 148 (1982), 193-214.  doi: 10.1007/BF02392728.  Google Scholar

[7]

J. Andersson, On the regularity of a free boundary near contact points with a fixed boundary, J. Differ. Equ., 232 (2007), 285-302.  doi: 10.1016/j.jde.2006.06.012.  Google Scholar

[8]

J. AnderssonN. Matevosyan and H. Mikayelyan, On the tangential touch between the free and the fixed boundaries for the two-phase obstacle-like problem, Ark. Mat., 44 (2006), 1-15.  doi: 10.1007/s11512-005-0005-2.  Google Scholar

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D. Arama and G. Leoni, On a variational approach for water waves, Commun. Partial Differ. Equ., 37 (2012), 833-874.  doi: 10.1080/03605302.2012.661819.  Google Scholar

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H. Berestycki, L. A. Caffarelli and L. Nirenberg, Uniform estimates for regularization of free boundary problems, in Analysis and Partial Differential Equations, vol. 122, Lecture Notes in Pure and Appl. Math., Dekker, New York, (1990), 567–619.  Google Scholar

[12]

A. Bonnet, On the regularity of edges in image segmentation, Ann. Inst. Henri Poincare Anal. Non Lineaire, 13 (1996), 485-528.  doi: 10.1016/S0294-1449(16)30111-1.  Google Scholar

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L. A. Caffarelli and A. Friedman, Axially symmetric infinite cavities, Indiana Univ. Math. J., 31 (1982), 135-160.  doi: 10.1512/iumj.1982.31.31014.  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, in Noncompact Problems at the Intersection of Geometry, Analysis, and Topology, vol. 350, Contemp. Math., Amer. Math. Soc., Providence, RI, (2004), 83–97. doi: 10.1090/conm/350/06339.  Google Scholar

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L. A. Caffarelli and S. Salsa, A Geometric Approach to Free Boundary Problems, vol. 68, Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2005. doi: 10.1090/gsm/068.  Google Scholar

[16]

H. Chang-Lara and O. Savin, Boundary regularity for the free boundary in the one-phase problem, in New developments in the analysis of nonlocal operators, vol. 723, Contemp. Math. doi: 10.1090/conm/723/14549.  Google Scholar

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A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. Math.(2), 173 (2011), 559-568.  doi: 10.4007/annals.2011.173.1.12.  Google Scholar

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G. Dal MasoJ. M. Morel and S. Solimini, A variational method in image segmentation: existence and approximation results, Acta Math., 168 (1992), 89-151.  doi: 10.1007/BF02392977.  Google Scholar

[19]

D. De Silva and D. Jerison, A singular energy minimizing free boundary, J. Reine Angew. Math., 635 (2009), 1-21.  doi: 10.1515/CRELLE.2009.074.  Google Scholar

[20]

N. Edelen and M. Engelstein, Quantitative stratification for some free-boundary problems, Trans. Amer. Math. Soc., 371 (2019), 2043-2072.  doi: 10.1090/tran/7401.  Google Scholar

[21]

A. Friedman, Variational Principles and Free-Boundary Problems, 2$^nd$ edition, Robert E. Krieger Publishing Co., Inc., Malabar, FL, 1988.  Google Scholar

[22]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2$^nd$ edition, Springer, New York, 1984. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[23]

G. Gravina and G. Leoni, On the existence of non-flat profiles for a Bernoulli free boundary problem, To appear, Adv. Calc. Var.. Google Scholar

[24]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Vol. 24, Monograph and Studies in Mathematics, Boston, Massachusetts, 1985.  Google Scholar

[25]

A. Gurevich, Boundary regularity for free boundary problems, Commun. Pure Appl. Math., 52 (1999), 363-403.  doi: 10.1002/(SICI)1097-0312(199903)52:3<363::AID-CPA3>3.3.CO;2-L.  Google Scholar

[26]

D. Jerison and O. Savin, Some remarks on stability of cones for the one-phase free boundary problem, Geom. Funct. Anal., 25 (2015), 1240-1257.  doi: 10.1007/s00039-015-0335-6.  Google Scholar

[27]

A. L. KarakhanyanC. E. Kenig and H. Shahgholian, The behavior of the free boundary near the fixed boundary for a minimization problem, Calc. Var. Partial Differ. Equ., 28 (2007), 15-31.  doi: 10.1007/s00526-006-0029-x.  Google Scholar

[28]

G. Keady and J. Norbury, On the existence theory for irrotational water waves, Math. Proc. Camb. Philos. Soc., 83 (1978), 137-157.  doi: 10.1017/S0305004100054372.  Google Scholar

[29]

J. P. Krasovskiǐ, On the theory of steady-state waves of finite amplitude, Ž. Vyčisl. Mat i Mat. Fiz., 1 (1961), 836–855.  Google Scholar

[30]

Y. Liu, Axially symmetric jet flows arising from high-speed fiber coating, Nonlinear Anal., 23 (1994), 319-363.  doi: 10.1016/0362-546X(94)90175-9.  Google Scholar

[31]

J. B. McLeod, The Stokes and Krasovskii conjectures for the wave of greatest height, Stud. Appl. Math., 98 (1997), 311-333.  doi: 10.1111/1467-9590.00051.  Google Scholar

[32]

P. I. Plotnikov, Proof of the Stokes conjecture in the theory of surface waves, Stud. Appl. Math., 108 (2002), 217–244. Translated from Dinamika Sploshn. Sredy No. 57 (1982), 41-76.  doi: 10.1111/1467-9590.01408.  Google Scholar

[33]

S. Raynor, Neumann fixed boundary regularity for an elliptic free boundary problem, Commun. Partial Differ. Equ., 33 (2008), 1975-1995.  doi: 10.1080/03605300802402658.  Google Scholar

[34]

J. F. Toland, On the existence of a wave of greatest height and Stokes's conjecture, Proc. R. Soc. Lond. A, 363 (1978), 469-485.  doi: 10.1098/rspa.1978.0178.  Google Scholar

[35]

E. Varvaruca and G. S. Weiss, A geometric approach to generalized Stokes conjectures, Acta Math., 206 (2011), 363-403.  doi: 10.1007/s11511-011-0066-y.  Google Scholar

[36]

E. Varvaruca and G. S. Weiss, The Stokes conjecture for waves with vorticity, Ann. Inst. Henri Poincare Anal. Non Lineaire, 29 (2012), 861-885.  doi: 10.1016/j.anihpc.2012.05.001.  Google Scholar

[37]

G. S. Weiss, Partial regularity for a minimum problem with free boundary, J. Geom. Anal., 9 (1999), 317-326.  doi: 10.1007/BF02921941.  Google Scholar

[38]

G. S. Weiss, Boundary monotonicity formulae and applications to free boundary problems. I. The elliptic case, Electron. J. Differ. Equ., (2004), Art. 44, 1–12.  Google Scholar

[39]

G. S. Weiss and G. Zhang, A free boundary approach to two-dimensional steady capillary gravity water waves, Arch. Ration. Mech. Anal., 203 (2012), 747-768.  doi: 10.1007/s00205-011-0466-3.  Google Scholar

[40]

G. S. Weiss and G. Zhang, The second variation of the stream function energy of water waves with vorticity, J. Differ. Equ., 253 (2012), 2646-2656.  doi: 10.1016/j.jde.2012.07.005.  Google Scholar

show all references

References:
[1]

H. W. Alt, Strömungen durch inhomogene poröse Medien mit freiem Rand, J. Reine Angew. Math., 305 (1979), 89-115.  doi: 10.1515/crll.1979.305.89.  Google Scholar

[2]

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.   Google Scholar

[3]

H. W. AltL. A. Caffarelli and A. Friedman, Axially symmetric jet flows, Arch. Rational Mech. Anal., 81 (1983), 97-149.  doi: 10.1007/BF00250648.  Google Scholar

[4]

H. W. AltL. A. Caffarelli and A. Friedman, Jets with two fluids. I. One free boundary, Indiana Univ. Math. J., 33 (1984), 213-247.  doi: 10.1512/iumj.1984.33.33011.  Google Scholar

[5]

C. J. Amick and L. E. Fraenkel, On the behavior near the crest of waves of extreme form, Trans. Amer. Math. Soc., 299 (1987), 273-298.  doi: 10.2307/2000494.  Google Scholar

[6]

C. J. AmickL. E. Fraenkel and J. F. Toland, On the Stokes conjecture for the wave of extreme form, Acta Math., 148 (1982), 193-214.  doi: 10.1007/BF02392728.  Google Scholar

[7]

J. Andersson, On the regularity of a free boundary near contact points with a fixed boundary, J. Differ. Equ., 232 (2007), 285-302.  doi: 10.1016/j.jde.2006.06.012.  Google Scholar

[8]

J. AnderssonN. Matevosyan and H. Mikayelyan, On the tangential touch between the free and the fixed boundaries for the two-phase obstacle-like problem, Ark. Mat., 44 (2006), 1-15.  doi: 10.1007/s11512-005-0005-2.  Google Scholar

[9]

D. Arama and G. Leoni, On a variational approach for water waves, Commun. Partial Differ. Equ., 37 (2012), 833-874.  doi: 10.1080/03605302.2012.661819.  Google Scholar

[10]

M. Bazarganzadeh and E. Lindgren, Tangential touch between the free and the fixed boundary in a semilinear free boundary problem in two dimensions, Ark. Mat., 52 (2014), 21-42.  doi: 10.1007/s11512-012-0179-3.  Google Scholar

[11]

H. Berestycki, L. A. Caffarelli and L. Nirenberg, Uniform estimates for regularization of free boundary problems, in Analysis and Partial Differential Equations, vol. 122, Lecture Notes in Pure and Appl. Math., Dekker, New York, (1990), 567–619.  Google Scholar

[12]

A. Bonnet, On the regularity of edges in image segmentation, Ann. Inst. Henri Poincare Anal. Non Lineaire, 13 (1996), 485-528.  doi: 10.1016/S0294-1449(16)30111-1.  Google Scholar

[13]

L. A. Caffarelli and A. Friedman, Axially symmetric infinite cavities, Indiana Univ. Math. J., 31 (1982), 135-160.  doi: 10.1512/iumj.1982.31.31014.  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, in Noncompact Problems at the Intersection of Geometry, Analysis, and Topology, vol. 350, Contemp. Math., Amer. Math. Soc., Providence, RI, (2004), 83–97. doi: 10.1090/conm/350/06339.  Google Scholar

[15]

L. A. Caffarelli and S. Salsa, A Geometric Approach to Free Boundary Problems, vol. 68, Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2005. doi: 10.1090/gsm/068.  Google Scholar

[16]

H. Chang-Lara and O. Savin, Boundary regularity for the free boundary in the one-phase problem, in New developments in the analysis of nonlocal operators, vol. 723, Contemp. Math. doi: 10.1090/conm/723/14549.  Google Scholar

[17]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. Math.(2), 173 (2011), 559-568.  doi: 10.4007/annals.2011.173.1.12.  Google Scholar

[18]

G. Dal MasoJ. M. Morel and S. Solimini, A variational method in image segmentation: existence and approximation results, Acta Math., 168 (1992), 89-151.  doi: 10.1007/BF02392977.  Google Scholar

[19]

D. De Silva and D. Jerison, A singular energy minimizing free boundary, J. Reine Angew. Math., 635 (2009), 1-21.  doi: 10.1515/CRELLE.2009.074.  Google Scholar

[20]

N. Edelen and M. Engelstein, Quantitative stratification for some free-boundary problems, Trans. Amer. Math. Soc., 371 (2019), 2043-2072.  doi: 10.1090/tran/7401.  Google Scholar

[21]

A. Friedman, Variational Principles and Free-Boundary Problems, 2$^nd$ edition, Robert E. Krieger Publishing Co., Inc., Malabar, FL, 1988.  Google Scholar

[22]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2$^nd$ edition, Springer, New York, 1984. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[23]

G. Gravina and G. Leoni, On the existence of non-flat profiles for a Bernoulli free boundary problem, To appear, Adv. Calc. Var.. Google Scholar

[24]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Vol. 24, Monograph and Studies in Mathematics, Boston, Massachusetts, 1985.  Google Scholar

[25]

A. Gurevich, Boundary regularity for free boundary problems, Commun. Pure Appl. Math., 52 (1999), 363-403.  doi: 10.1002/(SICI)1097-0312(199903)52:3<363::AID-CPA3>3.3.CO;2-L.  Google Scholar

[26]

D. Jerison and O. Savin, Some remarks on stability of cones for the one-phase free boundary problem, Geom. Funct. Anal., 25 (2015), 1240-1257.  doi: 10.1007/s00039-015-0335-6.  Google Scholar

[27]

A. L. KarakhanyanC. E. Kenig and H. Shahgholian, The behavior of the free boundary near the fixed boundary for a minimization problem, Calc. Var. Partial Differ. Equ., 28 (2007), 15-31.  doi: 10.1007/s00526-006-0029-x.  Google Scholar

[28]

G. Keady and J. Norbury, On the existence theory for irrotational water waves, Math. Proc. Camb. Philos. Soc., 83 (1978), 137-157.  doi: 10.1017/S0305004100054372.  Google Scholar

[29]

J. P. Krasovskiǐ, On the theory of steady-state waves of finite amplitude, Ž. Vyčisl. Mat i Mat. Fiz., 1 (1961), 836–855.  Google Scholar

[30]

Y. Liu, Axially symmetric jet flows arising from high-speed fiber coating, Nonlinear Anal., 23 (1994), 319-363.  doi: 10.1016/0362-546X(94)90175-9.  Google Scholar

[31]

J. B. McLeod, The Stokes and Krasovskii conjectures for the wave of greatest height, Stud. Appl. Math., 98 (1997), 311-333.  doi: 10.1111/1467-9590.00051.  Google Scholar

[32]

P. I. Plotnikov, Proof of the Stokes conjecture in the theory of surface waves, Stud. Appl. Math., 108 (2002), 217–244. Translated from Dinamika Sploshn. Sredy No. 57 (1982), 41-76.  doi: 10.1111/1467-9590.01408.  Google Scholar

[33]

S. Raynor, Neumann fixed boundary regularity for an elliptic free boundary problem, Commun. Partial Differ. Equ., 33 (2008), 1975-1995.  doi: 10.1080/03605300802402658.  Google Scholar

[34]

J. F. Toland, On the existence of a wave of greatest height and Stokes's conjecture, Proc. R. Soc. Lond. A, 363 (1978), 469-485.  doi: 10.1098/rspa.1978.0178.  Google Scholar

[35]

E. Varvaruca and G. S. Weiss, A geometric approach to generalized Stokes conjectures, Acta Math., 206 (2011), 363-403.  doi: 10.1007/s11511-011-0066-y.  Google Scholar

[36]

E. Varvaruca and G. S. Weiss, The Stokes conjecture for waves with vorticity, Ann. Inst. Henri Poincare Anal. Non Lineaire, 29 (2012), 861-885.  doi: 10.1016/j.anihpc.2012.05.001.  Google Scholar

[37]

G. S. Weiss, Partial regularity for a minimum problem with free boundary, J. Geom. Anal., 9 (1999), 317-326.  doi: 10.1007/BF02921941.  Google Scholar

[38]

G. S. Weiss, Boundary monotonicity formulae and applications to free boundary problems. I. The elliptic case, Electron. J. Differ. Equ., (2004), Art. 44, 1–12.  Google Scholar

[39]

G. S. Weiss and G. Zhang, A free boundary approach to two-dimensional steady capillary gravity water waves, Arch. Ration. Mech. Anal., 203 (2012), 747-768.  doi: 10.1007/s00205-011-0466-3.  Google Scholar

[40]

G. S. Weiss and G. Zhang, The second variation of the stream function energy of water waves with vorticity, J. Differ. Equ., 253 (2012), 2646-2656.  doi: 10.1016/j.jde.2012.07.005.  Google Scholar

Figure 1.  The free boundary meets the Dirichlet fixed boundary tangentially
Figure 2.  The free boundary hits the point $ (- \lambda/2, \gamma) $ at an angle of $ \pi/2 $
Figure 3.  Qualitative behavior of the free boundary near the fixed boundary
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