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

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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

Abstract
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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 and 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.
[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.
[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.
[4]	S. N. Bose, Plancks Gesetz und Lichtquantenhypothese, Zeitschrift für Physik, 26 (1924), 178-181. doi: 10.1007/BF01327326.
[5]	A. Braides, $\Gamma$-Convergence for Beginners, Oxford University Press, Oxford, 2002. doi: 10.1093/acprof:oso/9780198507840.001.0001.
[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.
[7]	G. Dal Maso, An Introduction to $\Gamma$-convergence, Birkhäuser, Basel, 1993. doi: 10.1007/978-1-4612-0327-8.
[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.
[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.
[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.
[11]	A. Einstein, Quantentheorie des einatomigen idealen Gases, Sitzungsberichte der Preussischen Akademie der Wissenschaften, (1925), 3-14. doi: 10.1002/3527608958.ch27.
[12]	L. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, American Mathematical Society, Providence, 1990.
[13]	L. Evans and R. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, 1992.
[14]	H. Federer, Geometric Measure Theory, Springer-Verlag, New York, 1969.
[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.
[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}.
[17]	E. P. Gross, Structure of a quantized vortex in boson systems, Nuovo Cimento, 20 (1961), 454-477. doi: 10.1007/BF02731494.
[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.
[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.
[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.
[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.
[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.
[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.
[24]	L. Modica, Gradient theory of phase transitions and minimal interface criterion, Arch. Rational Mech. Anal., 98 (1987), 123-142. doi: 10.1007/BF00251230.
[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.
[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.
[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.
[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.
[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.
[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.
[31]	L. P. Pitaevskii, Vortex lines in an imperfect Bose gas, Soviet Phys. JETP, 13 (1961), 451-454.
[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.
[33]	E. Timmermans, Phase separation of Bose-Einstein Condenstates, Phys. Rev. Let., 81 (1998), 5718-5721. doi: 10.1103/PhysRevLett.81.5718.

show all references

References:

[1]	A. Aftalion, Vortices in Bose-Einstein Condensates, Birkhäuser, Boston, 2006.
[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.
[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.
[4]	S. N. Bose, Plancks Gesetz und Lichtquantenhypothese, Zeitschrift für Physik, 26 (1924), 178-181. doi: 10.1007/BF01327326.
[5]	A. Braides, $\Gamma$-Convergence for Beginners, Oxford University Press, Oxford, 2002. doi: 10.1093/acprof:oso/9780198507840.001.0001.
[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.
[7]	G. Dal Maso, An Introduction to $\Gamma$-convergence, Birkhäuser, Basel, 1993. doi: 10.1007/978-1-4612-0327-8.
[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.
[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.
[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.
[11]	A. Einstein, Quantentheorie des einatomigen idealen Gases, Sitzungsberichte der Preussischen Akademie der Wissenschaften, (1925), 3-14. doi: 10.1002/3527608958.ch27.
[12]	L. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, American Mathematical Society, Providence, 1990.
[13]	L. Evans and R. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, 1992.
[14]	H. Federer, Geometric Measure Theory, Springer-Verlag, New York, 1969.
[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.
[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}.
[17]	E. P. Gross, Structure of a quantized vortex in boson systems, Nuovo Cimento, 20 (1961), 454-477. doi: 10.1007/BF02731494.
[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.
[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.
[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.
[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.
[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.
[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.
[24]	L. Modica, Gradient theory of phase transitions and minimal interface criterion, Arch. Rational Mech. Anal., 98 (1987), 123-142. doi: 10.1007/BF00251230.
[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.
[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.
[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.
[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.
[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.
[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.
[31]	L. P. Pitaevskii, Vortex lines in an imperfect Bose gas, Soviet Phys. JETP, 13 (1961), 451-454.
[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.
[33]	E. Timmermans, Phase separation of Bose-Einstein Condenstates, Phys. Rev. Let., 81 (1998), 5718-5721. doi: 10.1103/PhysRevLett.81.5718.

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