Abstract
We give a simple, new algebraic condition, directional control, which is sufficient for lower semicontinuity of surface energy and which is also very easy to check in practice, and we discuss and relate several other sufficient conditions. We establish an existence theorem for surface energy minimizers. We also show how to apply these results to minimal partitions, immiscible fluids (with and without gravity), soap bubble clusters, and curvature flow of polycrystals. In some cases, we use our results to give short, alternative proofs of important existence results in the literature. Our techniques are representative of those which could be used for many variational problems, both static and dynamic, involving interfaces. Our setting is that of the sets of finite perimeter and integral currents of geometric measure theory.
Similar content being viewed by others
References
Almgren, F.J. Jr.: Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints. Mem. Am. Math. Soc. 4(165), 199 (1976)
Almgren, F.J. Jr.: Deformations and multiple-valued functions, geometric measure theory and the calculus of variations. Proc. Symp. Pure Math. 44, 29–130 (1986)
Almgren, F.J. Jr.: Optimal isoperimetric inequalities. Indiana Univ. Math. J. 35, 451–547 (1986)
Almgren, F.J. Jr.: Questions and answers about area-minimizing surfaces and geometric measure theory. Proc. Symp. Pure Math. 54, 375–380 (1993)
Almgren, F.J. Jr., Taylor, J.E.: Soap bubble clusters. Forma 11(3), 199–207 (1996), reprinted in a book in The Kelvin Problem, Denis Weaire (ed.) Taylor and Francis. London pp. 37–45 (1996)
Almgren, F.J. Jr., Taylor, J.E., Wang, L.: Curvature driven flows: A variational approach. SIAM J. Control Optim. 31(2), 387–438 (1993)
Ambrosio, L., Braides, A.: Functionals defined on partitions in sets of finite perimeter I: Integral representation and Γ-convergence. J. Math. Pure. Appl. 69(9), 285–305 (1990)
Ambrosio, L., Braides, A.: Functionals defined on partitions in sets of finite perimeter II: Semicontinuity, relaxation and homogenization. J. Math. Pure. Appl. 69(9), 307–333 (1990)
Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, London (2000)
Braides, A.: Approximation of Free-Discontinuity Problems. Springer, Berlin (1998)
Brakke, K.A.: The Motion of a Surface by its Mean Curvature. Princeton University Press, Princeton (1978)
Burago, Y.D., Zalgaller, V.A.: Geometric Inequalities. Springer, New York (1988)
Caraballo, D.G.: A variational scheme for the evolution of polycrystals by curvature. Ph.D. thesis, Princeton University (1996)
Caraballo, D.G.: Existence and regularity of surface energy minimizing partitions of ℝn satisfying volume constraints (2007, in preparation)
Cheng, X.: A mass reducing flow for integral currents. Indiana Univ. Math. J. 42(2), 425–444 (1993)
Chung, K.Y.: On variational schemes modeling surface diffusion. Ph.D. thesis, Princeton University (1997)
Cook, E.A.: Free boundary regularity for surfaces minimizing Area(S)+cArea(∂ S). Trans. Am. Math. Soc. 290(2), 503–526 (1985)
Cook, E.A.: Su alcuni problemi comuni all’analisi e alla geometria. Note Mat. 9 (1989)
Dibos, F., Koepfler, G.: Global total variation minimization. SIAM J. Numer. Anal. 37(2), 646–664 (2000)
Ecker, K.: Area-minimizing integral currents with movable boundary parts of prescribed mass. Ann. Inst. H. Poincaré Anal. Non Linéaire 6(4), 261–293 (1989)
Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992)
Federer, H.: Geometric Measure Theory. Springer, Berlin (1969)
Federer, H., Fleming, W.H.: Normal and integral currents. Ann. Math. 72(2), 458–520 (1960)
Giusti, E.: Minimal Surfaces and Functions of Bounded Variation. Birkhäuser, Boston (1984)
Grayson, M.A.: Shortening embedded curves. Ann. Math. 129(2), 71–111 (1989)
Gurtin, M.E., Williams, W.O., Ziemer, W.P.: Geometric measure theory and the axioms of continuum thermodynamics. Arch. Ration. Mech. Anal. 92(1), 1–22 (1986)
Hardt, R., Simon, L.: Seminar on Geometric Measure Theory. Birkhäuser, Boston (1986)
Huisken, G.: Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom. 20, 237–266 (1984)
Krantz, S.G., Parks, H.R.: The Geometry of Domains in Space. Birkhäuser, Boston (1999)
Morgan, F.: Geometric Measure Theory: A Beginner’s Guide, 3rd edn. Academic Press, New York (2000)
Morgan, F.: Clusters minimizing area plus length of singular curves. Math. Ann. 299, 697–714 (1994)
Morgan, F.: Surfaces minimizing area plus length of singular curves. Proc. Am. Math. Soc. 122(4), 1153–1161 (1994)
Morgan, F.: Lowersemicontinuity of energy of clusters. Proc. R. Soc. Edinburgh Sect. A 127, 819–822 (1997)
Morgan, F.: Immiscible fluid clusters in R 2 and R 3. Mich. Math. J. 45(3), 441–450 (1998)
Morgan, F., Ritoré, M.: Geometric measure theory and the proof of the double bubble conjecture. Preprint
Morgan, F.: Personal communication
Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability. Cambridge University Press, Cambridge (1995)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Sethian, J.A.: Level set methods and fast marching methods. In: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science, 2nd edn. Cambridge Monogr. Appl. Comput. Math., vol. 3, p. 378. Cambridge University Press, Cambridge (1999)
Simon, L.: Lectures on Geometric Measure Theory. Centre for Mathematical Analysis, Australian National University (1984)
Sullivan, J.: Computing hypersurfaces which minimize surface energy plus bulk energy. In: Buttazzo, G., Visintin, A. (eds.) Motion by Mean Curvature and Related Topics, pp. 186–197. Walter de Gruyter, New York (1994)
Taylor, J.E.: Crystalline Variational Problems. Bull. Am. Math. Soc. (N.S.) 84(4), 568–588 (1978)
Taylor, J.E.: Some mathematical challenges in materials science. Bull. Am. Math. Soc. (N.S.) 40(1), 69–87 (2003)
Thompson, D’A.: On Growth and Form, abridged edn. Cambridge University Press, Cambridge (1961)
White, B.: Existence of least-energy configurations of immiscible fluids. J. Geom. Anal. 6, 151–161 (1996)
Yip, N.K.: Stochastic curvature driven flows. In: Stochastic Partial Differential Equations and Applications, Trento, 2002. Lecture Notes in Pure and Appl. Math., vol. 227, pp. 443–460. Dekker, New York (2002)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Caraballo, D.G. Crystals and Polycrystals in ℝn: Lower Semicontinuity and Existence. J Geom Anal 18, 68–88 (2008). https://doi.org/10.1007/s12220-007-9006-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12220-007-9006-7
Keywords
- Lower semicontinuity
- Partitions
- Clusters
- Polycrystals
- Compactness theorem
- Existence theorem
- Immiscible fluids