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BY 4.0 license Open Access Published by De Gruyter Open Access February 17, 2020

On the Surface Diffusion Flow with Triple Junctions in Higher Space Dimensions

H. Garcke and M. Gößwein
From the journal Geometric Flows

Abstract

We show short time existence for the evolution of triple junction clusters driven by the surface diffusion flow. On the triple line we use the boundary conditions derived by Garcke and Novick-Cohen as the singular limit of a Cahn-Hilliard equation with degenerated mobility. These conditions are concurrency of the triple junction, angle conditions between the hypersurfaces, continuity of the chemical potentials and a flux-balance. For the existence analysis we first write the geometric problem over a fixed reference surface and then use for the resulting analytic problem an approach in a parabolic Hölder setting.

MSC 2010: 53C44; 35K52; 35K93; 35R35; 35K55

References

[1] Abels, H., and Butz, J. Short time existence for the curve diffusion flow with a contact angle. J. Differential Equations 268, 1 (2019), 318–352.Search in Google Scholar

[2] Abels, H., Garcke, H., and Müller, L. Local well-posedness for volume-preserving mean curvature and Willmore flows with linear tension. Math. Nachr. 289, 2-3 (2016), 136–174.10.1002/mana.201400102Search in Google Scholar

[3] Agmon, S., Douglis, A., and Nirenberg, L. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I. Comm. Pure Appl. Math. 12 (1959), 623–727.Search in Google Scholar

[4] Barrett, J. W., Garcke, H., and Nürnberg, R. Parametric approximation of surface clusters driven by isotropic and anisotropic surface energies. Interfaces Free Bound. 12, 2 (2010), 187–234.Search in Google Scholar

[5] Bronsard, L., Garcke, H., and Stoth, B. A multi-phase Mullins-Sekerka system: matched asymptotic expansions and an implicit time discretisation for the geometric evolution problem. Proc. Roy. Soc. Edinburgh Sect. A 128, 3 (1998), 481–506.Search in Google Scholar

[6] Cahn, J. W., Elliott, C. M., and Novick-Cohen, A. The Cahn-Hilliard equation with a concentration dependent mobility: motion by minus the Laplacian of the mean curvature. European J. Appl. Math. 7, 3 (1996), 287–301.Search in Google Scholar

[7] Depner, D., and Garcke, H. Linearized stability analysis of surface diffusion for hypersurfaces with triple lines. Hokkaido Math. J. 42, 1 (2013), 11–52.Search in Google Scholar

[8] Depner, D., Garcke, H., and Kohsaka, Y. Mean curvature flow with triple junctions in higher space dimensions. Arch. Ration. Mech. Anal. 211, 1 (2014), 301–334.Search in Google Scholar

[9] Dziuk, G., Kuwert, E., and Schätzle, R. Evolution of elastic curves in ℝn: existence and computation. SIAM J. Math. Anal. 33, 5 (2002), 1228–1245.Search in Google Scholar

[10] Eidelman, S. D., and Zhitarashu, N. V. Parabolic boundary value problems, vol. 101 of Operator Theory: Advances and Applications. Birkhäuser Verlag, Basel, 1998.10.1007/978-3-0348-8767-0Search in Google Scholar

[11] Elliott, C. M., and Garcke, H. Existence results for diffusive surface motion laws. Adv. Math. Sci. Appl. 7, 1 (1997), 467–490.Search in Google Scholar

[12] Escher, J., Mayer, U. F., and Simonett, G. The surface diffusion flow for immersed hypersurfaces. SIAM J. Math. Anal. 29, 6 (1998), 1419–1433.Search in Google Scholar

[13] Evans, L. C. Partial differential equations, second ed., vol. 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2010.Search in Google Scholar

[14] Garcke, H., and Gößwein, M. Non-linear stability of double bubbles under surface diffusion. Preprint: arXiv:1910.01041 (2019).Search in Google Scholar

[15] Garcke, H., Ito, K., and Kohsaka, Y. Nonlinear stability of stationary solutions for surface diffusion with boundary conditions. SIAM J. Math. Anal. 40, 2 (2008), 491–515.Search in Google Scholar

[16] Garcke, H., Menzel, J., and Pluda, A. Willmore flow of planar networks. J. Differential Equations 266, 4 (2019), 2019–2051.Search in Google Scholar

[17] Garcke, H., and Novick-Cohen, A. A singular limit for a system of degenerate Cahn-Hilliard equations. Adv. Differential Equations 5, 4-6 (2000), 401–434.Search in Google Scholar

[18] Giga, Y., and Ito, K. On pinching of curves moved by surface diffusion. Commun. Appl. Anal. 2, 3 (1998), 393–405.Search in Google Scholar

[19] Giga, Y., and Ito, K. Loss of convexity of simple closed curves moved by surface diffusion. In Topics in nonlinear analysis, vol. 35 of Progr. Nonlinear Differential Equations Appl. Birkhäuser, Basel, 1999, pp. 305–320.10.1007/978-3-0348-8765-6_14Search in Google Scholar

[20] Gößwein, M. Surface diffusion flow of triple junction clusters in higher space dimensions. PhD thesis, Universität Regensburg, urn:nbn:de:bvb:355-epub-383760, 2019.Search in Google Scholar

[21] Kühnel, W. Differential geometry, vol. 77 of Student Mathematical Library. American Mathematical Society, Providence, RI, 2015.10.1090/stml/077Search in Google Scholar

[22] Ladyženskaja, O. A., Solonnikov, V. A., and Uralceva, N. N. Linear and quasilinear equations of parabolic type. Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23. American Mathematical Society, Providence, R.I., 1968.Search in Google Scholar

[23] Latushkin, Y., Prüß, J., and Schnaubelt, R. Stable and unstable manifolds for quasilinear parabolic systems with fully nonlinear boundary conditions. J. Evol. Equ. 6, 4 (2006), 537–576.Search in Google Scholar

[24] Lunardi, A. Analytic semigroups and optimal regularity in parabolic problems. Modern Birkhäuser Classics. Birkhäuser/Springer Basel AG, Basel, 1995.10.1007/978-3-0348-9234-6Search in Google Scholar

[25] Mullins, W. W. Theory of thermal grooving. J. Appl. Phys. 28, 3 (1957), 333–339.Search in Google Scholar

[26] Polden, A. Curves and surfaces of least total curvature and fourth-order flows. PhD thesis, Universität Thübingen, 1996.Search in Google Scholar

[27] Prüß, J., and Simonett, G. On the manifold of closed hypersurfaces in ℝn. Discrete Contin. Dyn. Syst. 33, 11-12 (2013), 5407–5428.10.3934/dcds.2013.33.5407Search in Google Scholar

[28] Spener, A. Short time existence for the elastic flow of clamped curves. Math. Nachr. 290, 13 (2017), 2052–2077.Search in Google Scholar

[29] Taylor, J. E., and Cahn, J. W. Linking anisotropic sharp and diffuse surface motion laws via gradient flows. J. Statist. Phys. 77, 1-2 (1994), 183–197.10.1007/BF02186838Search in Google Scholar

[30] Triebel, H. Theory of function spaces. II, vol. 84 of Monographs in Mathematics. Birkhäuser Verlag, Basel, 1992.10.1007/978-3-0346-0419-2Search in Google Scholar

Received: 2019-08-02
Accepted: 2020-01-13
Published Online: 2020-02-17

© 2020 H. Garcke et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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