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
Barles, G., Soner, H.M., Souganidis, P.E. (1993): Front propagation and phase field theory. SIAM J. Control Optim.,31, 439–469
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
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
Caginalp, G. (1986): An analysis of a phase-field model of a free boundary. Archive for Rational Mechanics and Analysis, 92, 205–245
Canarius, T., Schätzte, R. (1998): Finiteness and positivity results for global minimizers of a semilinear elliptic problem. Journal of Differential Equations, 148. 212–229
Canarius, T., Schätzte, R. (1999): Multiple solutions for a semilinear elliptic problem. To appear in Nonlinear Analysis TMA
Chen, X. (1992): Generation and Propagation of interface in reaction-diffusion equations. Journal of Differential Equations, 96, 116–141
Dancer, E.N. (1983): Breaking of Symmetries for Forced Equations. Mathematische Annalen, 262. 473–486
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
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
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
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
Friedman, A. (1968): The Stefan problem in several space variables. Transactions of the American Mathematical Society, 133. 51–87
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
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
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
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
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
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
Logak, E. (1996): Singular limit of reaction-diffusion systems and modified motion by mean curvature. preprint
Luckhaus, S. (1991): The Stefan Problem with Gibbs Thomson law. Sezione di Analisi Matematica e Probabilitita, Universita di Pisa, 2.75(591)
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
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
Reshetnyak, Y.G. (1968): Weak convergence of completely additive vector functions on a set. Siberian Mathematical Journal, 9, 1039–1045
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
Schätzle, R. (2000): The quasi-stationary phase field equations with Neumann boundary conditions. Journal of Differential Equations, 162. No.2, 473–503
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
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.
Souganidis, P.E., Soravia, P. (1996): Phase field theory for FitzHugh-Nagumo type systems. SIAM Journal on Mathematical Analysis, Vol. 27. 1341–1359
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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
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