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Geometric Conditions on Free Boundaries

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Geometric Analysis and Nonlinear Partial Differential Equations

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Summary

On free boundaries arising in models for melting and solidification of materials, conditions are often prescribed using geometrical data of the free boundary. In smooth models approximating these free boundary problems, the geometric conditions are replaced by elliptic and parabolic equations. We describe the approximation of mean curvature flow by the Allen- Cahn equation, also with coupling, and of the Stefan problem with Gibbs-Thomson law by the quasi-stationary phase field equations.

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References

  1. Barles, G., Soner, H.M., Souganidis, P.E. (1993): Front propagation and phase field theory. SIAM J. Control Optim.,31, 439–469

    Article  MathSciNet  MATH  Google Scholar 

  2. Barles, G., Souganidis, P.E. (1998): A New Approach to Front Propagation Problems: Theory and Applications. Archive for Rational Mechanics and Analysis, 141. 237–296

    Article  MathSciNet  MATH  Google Scholar 

  3. Bonami, A., Hilhorst, D., Logak, E. (2000): Modified motion by mean curvature: local existence and uniqueness and qualitative properties. Differential and Integral Equations, 13, no. 10-12, 1371–1392

    MathSciNet  MATH  Google Scholar 

  4. Caginalp, G. (1986): An analysis of a phase-field model of a free boundary. Archive for Rational Mechanics and Analysis, 92, 205–245

    Article  MathSciNet  MATH  Google Scholar 

  5. Canarius, T., Schätzte, R. (1998): Finiteness and positivity results for global minimizers of a semilinear elliptic problem. Journal of Differential Equations, 148. 212–229

    Article  MathSciNet  MATH  Google Scholar 

  6. Canarius, T., Schätzte, R. (1999): Multiple solutions for a semilinear elliptic problem. To appear in Nonlinear Analysis TMA

    Google Scholar 

  7. Chen, X. (1992): Generation and Propagation of interface in reaction-diffusion equations. Journal of Differential Equations, 96, 116–141

    Article  MathSciNet  MATH  Google Scholar 

  8. Dancer, E.N. (1983): Breaking of Symmetries for Forced Equations. Mathematische Annalen, 262. 473–486

    Article  MathSciNet  MATH  Google Scholar 

  9. Elliott, C.M., Paolini, M., Schätzte, R.: (1996): Interface estimates for the fully anisotropic Allen-Cahn equation and anisotropic mean-curvature flow. Math. Models Methods Appl. Sci., 6, No. 8, 1103–1118

    Article  MathSciNet  MATH  Google Scholar 

  10. Elliott, CM., Schätzte, R. (1996): The limit of the anisotropic double-obstacle Allen-Cahn equation. Proceedings of the Royal Society of Edinburgh, 126A. 1217–1234

    Article  Google Scholar 

  11. Elliott, CM., Schätzte, R. (1997): The limit of the fully anisotropic double-obstacle Allen-Cahn equation in the non-smooth case. SIAM Journal on Mathematical Analysis, Vol. 28, No. 2, 274–303

    Article  MathSciNet  MATH  Google Scholar 

  12. Evans, L.C, Soner, H.M., Souganidis, P.E. (1992): Phase transitions and generalized motion by mean curvature. Communications on Pure and Applied Mathematics, Vol. XLV, 1097–1123

    Article  MathSciNet  Google Scholar 

  13. Friedman, A. (1968): The Stefan problem in several space variables. Transactions of the American Mathematical Society, 133. 51–87

    Article  MathSciNet  MATH  Google Scholar 

  14. Giga, Y, Goto, S., Ishii, H. (1992): Global existence of weak solutions for interface equations coupled with diffusion equations. SIAM Journal on Mathematical Analysis, Vol.23, No. 4, 821–835

    Article  MathSciNet  MATH  Google Scholar 

  15. Gurtin, M., Matano, H. (1988): On the structure of equilibrium phase transitions within the gradient theory of fluids. Quarterly of Applied Mathematics, Vol. XLVI. No. 2, 301–317

    MathSciNet  Google Scholar 

  16. Henry, M., Hilhorst, D., Schätzte, R. (2000): Convergence to a viscosity solution for an advection-reaction-diffusion equation arising from a chemotaxis-growth model. Hiroshima Mathematical Journal

    Google Scholar 

  17. Hilhorst, D., Logak, E., Schätzte, R. (2000): Global existence for a nonlocal mean curvature flow as a limit of a parabolic-elliptic phase transition model. Interfaces and Free Boundaries, 2, 267–282

    Article  MathSciNet  MATH  Google Scholar 

  18. Hilhorst, D., Peletier, L.A., Schätzte, R. (2000): Γ-limit for the Extended Fisher-Kolmogorov equation. Preprint Universität Leiden No. MI 2000-16 May, to appear in Proceedings of the Royal Society of Edinburgh

    Google Scholar 

  19. Ilmanen, T. (1993): Convergence of the Allen-Cahn equation to Brakke’s motion by mean curvature. Journal of Differential Geometry, 38, No. 2, 417–461

    MathSciNet  MATH  Google Scholar 

  20. Logak, E. (1996): Singular limit of reaction-diffusion systems and modified motion by mean curvature. preprint

    Google Scholar 

  21. Luckhaus, S. (1991): The Stefan Problem with Gibbs Thomson law. Sezione di Analisi Matematica e Probabilitita, Universita di Pisa, 2.75(591)

    Google Scholar 

  22. Luckhaus, S., Modica, L. (1989): The Gibbs Thompson Relation within the Gradient Theory of Phase Transitions. Archive for Rational Mechanics and Analysis, 107. 71–83

    Article  MathSciNet  MATH  Google Scholar 

  23. Plotnikov, P.I., Starovoitov, V.N. (1993): Stefan Problem with Surface Tension as a Limit of the Phase Field Model. Differential Equations, 29. No.3, 395–404

    MathSciNet  Google Scholar 

  24. Reshetnyak, Y.G. (1968): Weak convergence of completely additive vector functions on a set. Siberian Mathematical Journal, 9, 1039–1045

    MATH  Google Scholar 

  25. Schätzle, R. (1997): A counterexample for an approximation of the Gibbs-Thomson law. Advances in Mathematical Sciences and Applications, Vol. 7, No. 1, 25–36

    MathSciNet  MATH  Google Scholar 

  26. Schätzle, R. (2000): The quasi-stationary phase field equations with Neumann boundary conditions. Journal of Differential Equations, 162. No.2, 473–503

    Article  MathSciNet  MATH  Google Scholar 

  27. Schätzle, R. (2001): Hypersurfaces with mean curvature given an ambient Sobolev function. Preprint Universität Freiburg Nr. 10/1999, habilitation thesis. To appear in Journal of Differential Geometry

    Google Scholar 

  28. Schätzle, R. (2002): A geometric regularity estimate via fully nonlinear ellitptic equations. International Mathematical Series, Vol. 1, Nonlinear Problems of Mathematical Physics and Related Topics, Volumes to honour Professor O.A.Ladyzenskaja to her 80th birthday.

    Google Scholar 

  29. Souganidis, P.E., Soravia, P. (1996): Phase field theory for FitzHugh-Nagumo type systems. SIAM Journal on Mathematical Analysis, Vol. 27. 1341–1359

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Mathematisches Institut, der Rheinischen Friedrich-Wilhelms-Universität Bonn, Beringstraße 6, D-53115, Bonn, Germany

    Reiner Schätzte

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  1. Reiner Schätzte
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Editors and Affiliations

  1. Department of Mathematics, University of Bonn, Beringstraße 1, 53115, Bonn, Germany

    Stefan Hildebrandt  & Hermann Karcher  & 

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© 2003 Springer-Verlag Berlin Heidelberg

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Schätzte, R. (2003). Geometric Conditions on Free Boundaries. In: Hildebrandt, S., Karcher, H. (eds) Geometric Analysis and Nonlinear Partial Differential Equations. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55627-2_13

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  • DOI: https://doi.org/10.1007/978-3-642-55627-2_13

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-44051-2

  • Online ISBN: 978-3-642-55627-2

  • eBook Packages: Springer Book Archive

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