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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 - A, 2016, 36 (5) : 2729-2755. doi: 10.3934/dcds.2016.36.2729
References:
[1]

A. Aftalion, Vortices in Bose-Einstein Condensates,, Birkhäuser, (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.  doi: 10.1007/s00526-014-0708-y.  Google Scholar

[3]

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

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

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

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G. Dal Maso, An Introduction to $\Gamma$-convergence,, Birkhäuser, (1993).  doi: 10.1007/978-1-4612-0327-8.  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.  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.  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, (1979), 131.   Google Scholar

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

[12]

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

[13]

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

[14]

H. Federer, Geometric Measure Theory,, Springer-Verlag, (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.  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.  doi: {10.1051/cocv/2014040}.  Google Scholar

[17]

E. P. Gross, Structure of a quantized vortex in boson systems,, Nuovo Cimento, 20 (1961), 454.  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, 3 (1987), 135.  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.  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.  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.  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.  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.  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.  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.   Google Scholar

[26]

G. Modugno, M. Modugno, F. Riboli and M. Inguscio, A two atomic species superfluid,, Phys. Rev. Lett., 89 (2002), 190404.  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).  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.  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.  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.  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.   Google Scholar

[32]

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

[33]

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

show all references

References:
[1]

A. Aftalion, Vortices in Bose-Einstein Condensates,, Birkhäuser, (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.  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.  doi: 10.1126/science.269.5221.198.  Google Scholar

[4]

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

[5]

A. Braides, $\Gamma$-Convergence for Beginners,, Oxford University Press, (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.  doi: 10.1515/CRELLE.2007.074.  Google Scholar

[7]

G. Dal Maso, An Introduction to $\Gamma$-convergence,, Birkhäuser, (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.  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.  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, (1979), 131.   Google Scholar

[11]

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

[12]

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

[13]

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

[14]

H. Federer, Geometric Measure Theory,, Springer-Verlag, (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.  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.  doi: {10.1051/cocv/2014040}.  Google Scholar

[17]

E. P. Gross, Structure of a quantized vortex in boson systems,, Nuovo Cimento, 20 (1961), 454.  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, 3 (1987), 135.  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.  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.  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.  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.  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.  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.  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.   Google Scholar

[26]

G. Modugno, M. Modugno, F. Riboli and M. Inguscio, A two atomic species superfluid,, Phys. Rev. Lett., 89 (2002), 190404.  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).  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.  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.  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.  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.   Google Scholar

[32]

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

[33]

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

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