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May  2016, 36(5): 2729-2755. doi: 10.3934/dcds.2016.36.2729

From gradient theory of phase transition to a generalized minimal interface problem with a contact energy

Tien-Tsan Shieh 1,

1. 

Department of Mathematics, National Central University, Chung-Li 320, Taiwan

Received  June 2015 Revised  July 2015 Published  October 2015

We consider asymptotic behaviours of a variational problem $$ \inf_{u\in \mathcal A(m,f)} \int_\Omega \left\{\frac{\epsilon^2}{2} \left|\nabla u\right|^2 + \frac{V(x)}{2}u^2 + \frac{1}{4}u^4\right\}\,dx$$ over a admissible class $\mathcal A(m,f)=\{u\in W^{1,2}(\Omega):\,\int_\Omega u^2\,dx=m,\,u=f \textrm{ on }\partial \Omega\}$. The problem demonstrates some features of the phase separation in experimental studies of Bose-Einstein condensation confined in an infinite-trap potential. In this paper, we show the limiting variational problem is a generalized minimal interface problem involving a boundary contact energy. The asymptotic behaviour of the minimizers $\{u_\epsilon\}$ is characterized by a generalized mean curvature equation and a contact angle relation, the Young's relation, at the junction of the interfaces and the boundary. An example is given to demonstrate the possible existence of local minimizers $\{v_\epsilon\}_{\epsilon>0}$ for the perturbed variational problem due to suitable Dirichlet boundary condition $u=f$.
Keywords: minimal interfaces, contact angles., Cahn-Hilliard equation, $\Gamma$-convergence, sharp-interface limit, local minimizers.
Mathematics Subject Classification: Primary: 49J45, 35J25, 35J75; Secondary: 35Q40, 49S0.
Citation: Tien-Tsan Shieh. From gradient theory of phase transition to a generalized minimal interface problem with a contact energy. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2729-2755. doi: 10.3934/dcds.2016.36.2729
References:
[1]

A. Aftalion, Vortices in Bose-Einstein Condensates, Birkhäuser, Boston, 2006.  Google Scholar

[2]

A. Aftalion and J. Royo-Letelier, A mimimal interface problem arising from a two component Bose-Einstein condensate via $\Gamma$- convergence, Calc. Var. Partial Differential Equations, 52 (2015), 165-197. doi: 10.1007/s00526-014-0708-y.  Google Scholar

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S. N. Bose, Plancks Gesetz und Lichtquantenhypothese, Zeitschrift für Physik, 26 (1924), 178-181. doi: 10.1007/BF01327326.  Google Scholar

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A. Braides, $\Gamma$-Convergence for Beginners, Oxford University Press, Oxford, 2002. doi: 10.1093/acprof:oso/9780198507840.001.0001.  Google Scholar

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F. Dalfovo, S. Giorgini, L. P. Pitaevskii and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys., 71 (1999), 463-512. doi: 10.1103/RevModPhys.71.463.  Google Scholar

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K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. D. Durfee, D. M. Kurn and W. Ketterle, Bose-Einstein condensation in a gas of sodium atoms, Phys. Rev. Lett., 75 (1995), 3969-3973. doi: 10.1103/PhysRevLett.75.3969.  Google Scholar

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E. De Giorgi, Convergence problems for functionals and operators, in Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (ed. E. Magenes, U. Mosco and E. De Giorgi), Pitagoria Ed. Bologna, 1979, 131-188.  Google Scholar

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L. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, American Mathematical Society, Providence, 1990.  Google Scholar

[13]

L. Evans and R. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, 1992.  Google Scholar

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H. Federer, Geometric Measure Theory, Springer-Verlag, New York, 1969.  Google Scholar

[15]

I. Fonseca and L. Tartar, The gradient theory of phase transitions for systems with two potential wells, Proc. Roy. Soc. Edingburgh Sect. A, 111 (1989), 89-102. doi: 10.1017/S030821050002504X.  Google Scholar

[16]

M. Goldman and J. Royo-Letelier, Sharp interface limit for two components Bose-Einstein condensates, ESAIM Control Optim. Calc. Var., 21 (2015), 603-624. doi: {10.1051/cocv/2014040}.  Google Scholar

[17]

E. P. Gross, Structure of a quantized vortex in boson systems, Nuovo Cimento, 20 (1961), 454-477. doi: 10.1007/BF02731494.  Google Scholar

[18]

M. Gurtin, Some results and conjectures in the gradient theory of phase transitions, in Metastability and incompletely posed problems (ed. Stuart, Atman, Ericksen, Kinderlehrer and Mülcer), Springer-Verlag, 3 (1987), 135-146. doi: 10.1007/978-1-4613-8704-6_9.  Google Scholar

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D. Hall, M. Matthews, C. Wieman and E. Cornell, Measurements of relative phase in binary mixtures of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), 1543-1547. doi: 10.1103/PhysRevLett.81.1543.  Google Scholar

[20]

K. Ishige, Singular perturbations of variational problems of vector valued functions, Nonlinear Anal., 23 (1994), 1453-1466. doi: 10.1016/0362-546X(94)90139-2.  Google Scholar

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K. Ishige, The gradient theory of the phase transitions in Cahn-Hilliard fluids with the Dirichlet boundary conditions, SIAM J. Math. Anal., 27 (1996), 620-637. doi: 10.1137/0527034.  Google Scholar

[22]

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[23]

F. Lin, X. B. Pan and C. Y. Wang, Phase transition for potentials of high-dimensional wells, Cumm. Pure Appl. Math., 65 (2012), 833-888. doi: 10.1002/cpa.21386.  Google Scholar

[24]

L. Modica, Gradient theory of phase transitions and minimal interface criterion, Arch. Rational Mech. Anal., 98 (1987), 123-142. doi: 10.1007/BF00251230.  Google Scholar

[25]

L. Modica, Gradient theory of phase transitions with boundary contact energy, Ann Inst. H. Poincar'e Anal. Non Lin'eaire, 4 (1987), 487-512.  Google Scholar

[26]

G. Modugno, M. Modugno, F. Riboli and M. Inguscio, A two atomic species superfluid, Phys. Rev. Lett., 89 (2002), 190404-190408. doi: 10.1103/PhysRevLett.89.190404.  Google Scholar

[27]

R. Navarro, R. Carretero-González and P. G. Kevrekidis, Phase separation and dynamics of two-component Bose-Einstein condensates, Phy. Rev. A, 80 (2009), 023613. doi: 10.1103/PhysRevA.80.023613.  Google Scholar

[28]

N. Owen, Nonconvex variational problems with general singular perturbations, Trans. Am. math. Soc., 310 (1988), 393-404. doi: 10.1090/S0002-9947-1988-0965760-9.  Google Scholar

[29]

N. Owen and P. Sternberg, Nonconvex variational problems with anisotropic perturbations, Nonlinear Anal., 16 (1991), 705-719. doi: 10.1016/0362-546X(91)90177-3.  Google Scholar

[30]

N. Owen, J. Rubinstein and P. Sternberg, Minimizers and gradient flows for singularly perturbed bi-stable potentials with a Dirichlet condition, Proc. Roy. Soc. London Ser. A, 429 (1990), 505-532. doi: 10.1098/rspa.1990.0071.  Google Scholar

[31]

L. P. Pitaevskii, Vortex lines in an imperfect Bose gas, Soviet Phys. JETP, 13 (1961), 451-454. Google Scholar

[32]

P. Sternberg, The effect of a singular perturbation on nonconvex variational problems, Arch. Rational Mech. Anal., 101 (1988), 209-260. doi: 10.1007/BF00253122.  Google Scholar

[33]

E. Timmermans, Phase separation of Bose-Einstein Condenstates, Phys. Rev. Let., 81 (1998), 5718-5721. doi: 10.1103/PhysRevLett.81.5718.  Google Scholar

show all references

References:
[1]

A. Aftalion, Vortices in Bose-Einstein Condensates, Birkhäuser, Boston, 2006.  Google Scholar

[2]

A. Aftalion and J. Royo-Letelier, A mimimal interface problem arising from a two component Bose-Einstein condensate via $\Gamma$- convergence, Calc. Var. Partial Differential Equations, 52 (2015), 165-197. doi: 10.1007/s00526-014-0708-y.  Google Scholar

[3]

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman and E. A. Cornell, Observation of Bose-Einstein condensation in a dilute atomic vapor, Science, 269 (1995), 198-201. doi: 10.1126/science.269.5221.198.  Google Scholar

[4]

S. N. Bose, Plancks Gesetz und Lichtquantenhypothese, Zeitschrift für Physik, 26 (1924), 178-181. doi: 10.1007/BF01327326.  Google Scholar

[5]

A. Braides, $\Gamma$-Convergence for Beginners, Oxford University Press, Oxford, 2002. doi: 10.1093/acprof:oso/9780198507840.001.0001.  Google Scholar

[6]

R. Choksi and P. Sternberg, On the first and second variations of a nonlocal isoperimetric problem, J. Reine Angew. Math., 611 (2007), 75-108. doi: 10.1515/CRELLE.2007.074.  Google Scholar

[7]

G. Dal Maso, An Introduction to $\Gamma$-convergence, Birkhäuser, Basel, 1993. doi: 10.1007/978-1-4612-0327-8.  Google Scholar

[8]

F. Dalfovo, S. Giorgini, L. P. Pitaevskii and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys., 71 (1999), 463-512. doi: 10.1103/RevModPhys.71.463.  Google Scholar

[9]

K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. D. Durfee, D. M. Kurn and W. Ketterle, Bose-Einstein condensation in a gas of sodium atoms, Phys. Rev. Lett., 75 (1995), 3969-3973. doi: 10.1103/PhysRevLett.75.3969.  Google Scholar

[10]

E. De Giorgi, Convergence problems for functionals and operators, in Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (ed. E. Magenes, U. Mosco and E. De Giorgi), Pitagoria Ed. Bologna, 1979, 131-188.  Google Scholar

[11]

A. Einstein, Quantentheorie des einatomigen idealen Gases, Sitzungsberichte der Preussischen Akademie der Wissenschaften, (1925), 3-14. doi: 10.1002/3527608958.ch27.  Google Scholar

[12]

L. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, American Mathematical Society, Providence, 1990.  Google Scholar

[13]

L. Evans and R. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, 1992.  Google Scholar

[14]

H. Federer, Geometric Measure Theory, Springer-Verlag, New York, 1969.  Google Scholar

[15]

I. Fonseca and L. Tartar, The gradient theory of phase transitions for systems with two potential wells, Proc. Roy. Soc. Edingburgh Sect. A, 111 (1989), 89-102. doi: 10.1017/S030821050002504X.  Google Scholar

[16]

M. Goldman and J. Royo-Letelier, Sharp interface limit for two components Bose-Einstein condensates, ESAIM Control Optim. Calc. Var., 21 (2015), 603-624. doi: {10.1051/cocv/2014040}.  Google Scholar

[17]

E. P. Gross, Structure of a quantized vortex in boson systems, Nuovo Cimento, 20 (1961), 454-477. doi: 10.1007/BF02731494.  Google Scholar

[18]

M. Gurtin, Some results and conjectures in the gradient theory of phase transitions, in Metastability and incompletely posed problems (ed. Stuart, Atman, Ericksen, Kinderlehrer and Mülcer), Springer-Verlag, 3 (1987), 135-146. doi: 10.1007/978-1-4613-8704-6_9.  Google Scholar

[19]

D. Hall, M. Matthews, C. Wieman and E. Cornell, Measurements of relative phase in binary mixtures of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), 1543-1547. doi: 10.1103/PhysRevLett.81.1543.  Google Scholar

[20]

K. Ishige, Singular perturbations of variational problems of vector valued functions, Nonlinear Anal., 23 (1994), 1453-1466. doi: 10.1016/0362-546X(94)90139-2.  Google Scholar

[21]

K. Ishige, The gradient theory of the phase transitions in Cahn-Hilliard fluids with the Dirichlet boundary conditions, SIAM J. Math. Anal., 27 (1996), 620-637. doi: 10.1137/0527034.  Google Scholar

[22]

R. Kohn and P. Sternberg, Local minimiser and singular perturbations, Proc. Roy. Soc. Edinburg Sect. A, 111 (1989), 69-84. doi: 10.1017/S0308210500025026.  Google Scholar

[23]

F. Lin, X. B. Pan and C. Y. Wang, Phase transition for potentials of high-dimensional wells, Cumm. Pure Appl. Math., 65 (2012), 833-888. doi: 10.1002/cpa.21386.  Google Scholar

[24]

L. Modica, Gradient theory of phase transitions and minimal interface criterion, Arch. Rational Mech. Anal., 98 (1987), 123-142. doi: 10.1007/BF00251230.  Google Scholar

[25]

L. Modica, Gradient theory of phase transitions with boundary contact energy, Ann Inst. H. Poincar'e Anal. Non Lin'eaire, 4 (1987), 487-512.  Google Scholar

[26]

G. Modugno, M. Modugno, F. Riboli and M. Inguscio, A two atomic species superfluid, Phys. Rev. Lett., 89 (2002), 190404-190408. doi: 10.1103/PhysRevLett.89.190404.  Google Scholar

[27]

R. Navarro, R. Carretero-González and P. G. Kevrekidis, Phase separation and dynamics of two-component Bose-Einstein condensates, Phy. Rev. A, 80 (2009), 023613. doi: 10.1103/PhysRevA.80.023613.  Google Scholar

[28]

N. Owen, Nonconvex variational problems with general singular perturbations, Trans. Am. math. Soc., 310 (1988), 393-404. doi: 10.1090/S0002-9947-1988-0965760-9.  Google Scholar

[29]

N. Owen and P. Sternberg, Nonconvex variational problems with anisotropic perturbations, Nonlinear Anal., 16 (1991), 705-719. doi: 10.1016/0362-546X(91)90177-3.  Google Scholar

[30]

N. Owen, J. Rubinstein and P. Sternberg, Minimizers and gradient flows for singularly perturbed bi-stable potentials with a Dirichlet condition, Proc. Roy. Soc. London Ser. A, 429 (1990), 505-532. doi: 10.1098/rspa.1990.0071.  Google Scholar

[31]

L. P. Pitaevskii, Vortex lines in an imperfect Bose gas, Soviet Phys. JETP, 13 (1961), 451-454. Google Scholar

[32]

P. Sternberg, The effect of a singular perturbation on nonconvex variational problems, Arch. Rational Mech. Anal., 101 (1988), 209-260. doi: 10.1007/BF00253122.  Google Scholar

[33]

E. Timmermans, Phase separation of Bose-Einstein Condenstates, Phys. Rev. Let., 81 (1998), 5718-5721. doi: 10.1103/PhysRevLett.81.5718.  Google Scholar

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