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On the structure of phase transition maps for three or more coexisting phases

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Geometric Partial Differential Equations proceedings

Part of the book series: CRM Series ((CRMSNS,volume 15))

Abstract

We consider system (1.1) below. After stating certain general facts which do not depend on the structure of W, we focus on the phase transition case. For symmetric W’s the main issues have been resolved, and we summarize them here. Next we recall the De Giorgi conjecture in the scalar case, and we point to a Bernstein type theorem that appears appropriate for (1.1). Finally we state a result on the hierarchical structure of the equivariant solutions.

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References

  1. G. Alberti, Variational methods for phase transitions, an approach via Γ-convergence, In: L. Ambrosio and N. Dancer, “Calculus of Variations and Partial Differential Equations”, G. Buttazzo, A. Marino, and M. K. V. Murthy (eds.) Springer, Berlin, 2000, 95–114.

    Chapter  Google Scholar 

  2. G. Alberti, L. Ambosio and X. Cabré, On a long-standing conjecture of E. De Giorgi: Symmetry in 3D for general nonlinearities and a local minimality property, Acta Appl. Math. 65 (2001), 9–33.

    Article  MATH  MathSciNet  Google Scholar 

  3. N. D. Alikakos, Some basic facts on the system Δu − Wu(u) = 0, Proc. Amer. Math. Soc. 139 (2011), 153–162.

    Article  MATH  MathSciNet  Google Scholar 

  4. N. D. Alikakos, A new proof for the existence of an equivariant entire solution connecting the minima of the potential for the system Δu − Wu(u) = 0, Comm. Partial Diff. Eqs. 37 (2012), 2093–2115.

    Article  MATH  MathSciNet  Google Scholar 

  5. N. D. Alikakos, P. Antonopoulos and A. Damialis, Plateau angle conditions for the vector-valued Allen—Cahn equation, preprint.

    Google Scholar 

  6. N. D. Alikakos and P. W. Bates, in preparation.

    Google Scholar 

  7. N. D. Alikakos and A. C. Faliagas, The stress-energy tensorand Pohozaev’s identity for systems, Acta Math. Scientia 32 (2012), 433–439.

    Article  MATH  MathSciNet  Google Scholar 

  8. N. D. Alikakos and A. C. Faliagas, in preparation.

    Google Scholar 

  9. N. D. Alikakos and G. Fusco, Entire solutions to nonconvex variational elliptic systems in the presence of a finite symmetry group, In: “Singularities in Nonlinear Evolution Phenomena and Applications”, M. Novaga and G. Orlandi (eds), Publications of the Scuola Normale Superiore, CRM Series, Birkhäuser, 2009, 1–26.

    Google Scholar 

  10. N. D. Alikakos and G. Fusco, Entire solutions to equivariant elliptic systems with variational structure, Arch. Rat. Mech. Anal. 202 (2011), 567–597.

    Article  MATH  MathSciNet  Google Scholar 

  11. N. D. Alikakos and G. Fusco, On the asymptotic behavior of symmetric solutions of the elliptic system Δu = Wu(u) in unbounded domains, preprint.

    Google Scholar 

  12. F. J. Almgren, JR.., “Existence and Regularity Almost Everywhere of Solutions to Elliptic Variational Problems with Constraints”, Mem. Amer. Math. Soc. 165, 1976.

    Google Scholar 

  13. L. Ambrosio and X. Cabré, Entire solutions of semilinear elliptic equations in ℝ3 and a conjecture of De Giorgi, J. Amer. Math. Soc. 13 (2000), 725–739.

    Article  MATH  MathSciNet  Google Scholar 

  14. S. Baldo, Minimal interface criterion for phase transitions in mixtures of Cahn—Hilliard fluids, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 7 (1990), 67–90.

    MATH  MathSciNet  Google Scholar 

  15. P. W. Bates, G. Fusco and P. Smyrnelis, Crystalline and other entire solutions of the vector Allen—Cahn equation, preprint.

    Google Scholar 

  16. H. Berestycki, T.-C. Lin, J. Wei and C. Zhao, On phase-separation models: Asymptotics and qualitative properties, Arch. Rat. Mech. Anal. 208 (2013), 163–200.

    Article  MATH  MathSciNet  Google Scholar 

  17. H. Berestycki, S. Terracini, K. Wang and J. Wei, Existence and stability of entire solutions of an elliptic system modeling phase separation, Adv. Math., to appear.

    Google Scholar 

  18. F. Bethuel, H. Brezis and F. Hélein, “Ginzburg—Landau Vortices”, Progress in Nonlinear Differential Equations and Their Applications, Vol. 13, Birkhäuser, Basel and Boston, 1994.

    Google Scholar 

  19. L. Bronsard, C. Gui and M. Schatzman, A three-layered minimizer in R2 for a variational problem with a symmetric three-well potential, Comm. Pure Appl. Math. 49 (1996), 677–715.

    Article  MATH  MathSciNet  Google Scholar 

  20. L. Caffarelli, N. Garofalo and F. Segala, A gradient bound for entire solutions of quasi-linear equations and its consequences, Comm. Pure Appl. Math. 47 (1994), 1457–1473.

    Article  MATH  MathSciNet  Google Scholar 

  21. C. C. Chan, “Structure of the Singular set in Energy-minimizing Partitions and Area-minimizing Surfaces in RN”, Ph.D. thesis, Stanford University, 1995.

    Google Scholar 

  22. T. H. Colding and W. P. Minicozzi II, “A Course in Minimal Surfaces”, Graduate Studies in Mathematics, Vol. 121, American Mathematical Society, Providence, RI, 2011.

    Google Scholar 

  23. E. De Giorgi, Convergence problems for functional and operators, In: E. De Giorgi, E. Magenes, and U. Mosco, “Recent methods in non-linear analysis”, Proc. Int. Meet., Rome, 1978. Pitagora, Bologna, 1979, 131–188.

    Google Scholar 

  24. M. del Pino, M. Kowalczyk and J. Wei, On De Giorgi’s conjecture in dimension N ≥ 9, Ann. Math. (2) 174 (2011), 1485–1569.

    Article  MATH  MathSciNet  Google Scholar 

  25. M. del Pino, M. Kowalczyk and J. Wei On De Giorgi’s conjecture and beyond, Proc. Natl. Acad. Sci. USA 109 (2012), 6845–6850.

    Article  MATH  MathSciNet  Google Scholar 

  26. L. C. Evans, “Partial Differential Equations”, Graduate Studies in Mathematics 19, second edition, American Mathematical Society, Providence, RI, 2010.

    Google Scholar 

  27. A. Farina, Two results on entire solutions of Ginzburg—Landau system in higher dimensions, J. Funct. Anal. 214 (2004), 386–395.

    Article  MATH  MathSciNet  Google Scholar 

  28. A. Farina, Some symmetry results for entire solutions of an elliptic system arising in phase separation, Discrete Contin. Dyn. Syst., Ser. A, to appear.

    Google Scholar 

  29. A. Farina and N. Soave, Monotonicity and 1-dimensional symmetry for solutions of an elliptic system arising in Bose—Einstein condensation, preprint.

    Google Scholar 

  30. A. Farina and E. Valdinoci, The state of the art for a conjecture of de Giorgi and related problems, In: “Recent Progress on Reaction-diffusion Systems and Viscosity Solutions”, H. Ishii, W.-Y. Lin and Y. Du (eds.), World Scientific, Hackensack, NJ, 2009, 74–96.

    Chapter  Google Scholar 

  31. M. Fazly and N. Ghoussoub, De Giorgi type results for elliptic systems, Calc. Var., in press.

    Google Scholar 

  32. W. H. Fleming, On the oriented Plateau problem, Rend. Circ. Mat. Palermo 11 (1962), 69–90.

    Article  MATH  MathSciNet  Google Scholar 

  33. G. Fusco, Equivariant entire solutions to the elliptic system Δu = Wu(u) for general G-invariant potentials, preprint.

    Google Scholar 

  34. N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems, Ann. Math. (2) 157 (2003), 313–334.

    Article  MATH  MathSciNet  Google Scholar 

  35. L. C. Grove and C. T. Benson, “Finite Reflection Groups”, Graduate Texts in Mathematics, Vol. 99, second edition. Springer-Verlag, Berlin, 1985.

    Google Scholar 

  36. C. Gui, Hamiltonian identities for elliptic differential equations, J. Funct. Anal. 254 (2008), 904–933.

    Article  MATH  MathSciNet  Google Scholar 

  37. C. Gui and M. Schatzman, Symmetric quadruple phase transitions, Indiana Univ. Math. J. 57 (2008), 781–836.

    Article  MATH  MathSciNet  Google Scholar 

  38. D. Kinderlehrer, L. Nirenberg and J. Spruck, Regularity in elliptic free boundary problems I, J. Anal. Math. 34 (1978), 86–119.

    Article  MATH  MathSciNet  Google Scholar 

  39. M. Kowalczyk, Y. Liu and F. Pacard, The classification of four-end solutions to the Allen—Cahn equation on the plane, Anal. PDE, to appear.

    Google Scholar 

  40. L. D. Landau and E. M. Lifshitz, “The Classical Theory of Fields”, Course of Theoretical Physics 2, fourth edition, Butterworth-Heinemann, 1975.

    Google Scholar 

  41. L. Modica, A gradient bound and a Liouville theorem for nonlinear Poisson equations, Comm. Pure Appl. Math. 38 (1985), 679–684.

    Article  MATH  MathSciNet  Google Scholar 

  42. L. Modica, Monotonicity of the energy for entire solutions of semilinear elliptic equations, In: “Partial Differential Equations and the Calculus of Variations”, F. Colombini, A. Marino, L. Modica and S. Spagnolo (eds.), Vol. II, Essays in honor of Ennio De Giorgi, Progress in Nonlinear Differential Equations and Their Applications, Vol. 2, Birkhäuser, Boston, MA, 1989, 843–850.

    Chapter  Google Scholar 

  43. F. Morgan, Harnack-type mass bounds and Bernstein theorems for area-minimizing flat chains modulo ν, Comm. Partial Diff. Eqs. 11 (1986), 1257–1283.

    Article  MATH  Google Scholar 

  44. O. Savin, Regularity of flat level sets in phase transitions, Ann. Math. (2) 169 (2009), 41–78.

    Article  MATH  MathSciNet  Google Scholar 

  45. L. Simon, “Lectures on Geometric Measure Theory”, Proc. Centre Math. Anal. 3. Australian National University, Canberra, 1983.

    Google Scholar 

  46. L. Simon, Cylindrical tangent cones and the singular set of minimal submanifolds, J. Diff. Geom. 38 (1993), 585–652.

    MATH  Google Scholar 

  47. M. Spivak, “A Comprehensive Introduction to Differential Geometry”, Vol. 4, third edition, Publish or Perish, Houston, TX, 1999.

    Google Scholar 

  48. V. Stefanopoulos, Heteroclinic connections for multiple-well potentials: The anisotropic case, Proc. R. Soc. Edinb. A 138 (2008), 1313–1330.

    Article  MATH  MathSciNet  Google Scholar 

  49. P. Sternberg and K. Zumbrun, A Poincaré inequality with applications to volume-constrained area-minimizing surfaces, J. Reine Angew. Math. 503 (1998), 63–85.

    MATH  MathSciNet  Google Scholar 

  50. J. Taylor, The structure of singularities in soap-bubble and soap-bubble-like minimal surfaces, Ann. Math. (2) 103 (1976), 489–539.

    Article  MATH  Google Scholar 

  51. B. White, Existence of least-energy configurations of immiscible fluids, J. Geom. Anal. 6 (1996), 151–161.

    Article  MATH  MathSciNet  Google Scholar 

  52. B. White, “Topics in Geometric Measure Theory”, Lecture notes, Stanford University, 2012.

    Google Scholar 

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Authors
  1. Nicholas D. Alikakos
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Editors and Affiliations

  1. CMAP, Ecole Polytechnique, CNRS, 91128, Palaiseau Cedex, France

    Antonin Chambolle

  2. Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127, Pisa, Italia

    Matteo Novaga

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Alikakos, N.D. (2013). On the structure of phase transition maps for three or more coexisting phases. In: Chambolle, A., Novaga, M., Valdinoci, E. (eds) Geometric Partial Differential Equations proceedings. CRM Series, vol 15. Edizioni della Normale, Pisa. https://doi.org/10.1007/978-88-7642-473-1_1

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