A note on non lower semicontinuous perimeter functionals on partitions

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September 2016, 11(3): 501-508. doi: 10.3934/nhm.2016006

A note on non lower semicontinuous perimeter functionals on partitions

Annibale Magni ^1, and Matteo Novaga ^2,

1.	Westfälische-Wilhelms Universität Münster, Einsteinstrasse 62, 48149 Münster, Germany
2.	Dipartimento di Matematica, Universitá di Pisa, Largo Bruno Pontecorvo 5, I-56127 Pisa

Received May 2015 Revised August 2015 Published August 2016

Abstract
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We consider isotropic non lower semicontinuous weighted perimeter functionals defined on partitions of domains in $\mathbb{R}^n$. Besides identifying a condition on the structure of the domain which ensures the existence of minimizing configurations, we describe the structure of such minima, as well as their regularity.

Keywords: regularity of interfaces., Semicontinuity, minimal partitions.

Mathematics Subject Classification: Primary: 49Q20; Secondary: 76M30, 76T3.

Citation: Annibale Magni, Matteo Novaga. A note on non lower semicontinuous perimeter functionals on partitions. Networks and Heterogeneous Media, 2016, 11 (3) : 501-508. doi: 10.3934/nhm.2016006

References:

[1]	L. Ambrosio and A. Braides, Functionals defined on partitions in sets of finite perimeter. II, J. Math. Pures Appl., 69 (1990), 307-333.
[2]	E. Bretin and S. Masnou, A new phase field model for inhomogeneous minimal partitions, and applications to droplets dynamics, Preprint, 2015.
[3]	S. Esedoglu and F. Otto, Threshold dynamics for networks with arbitrary surface tensions, Comm. Pure Appl. Math., 68 (2015), 808-864. doi: 10.1002/cpa.21527.
[4]	G. P. Leonardi, Infiltrations in immiscible fluids systems, Proc. Roy Soc. Edimburgh, 131 (2001), 425-436. doi: 10.1017/S0308210500000937.
[5]	F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems, Cambridge studies in advanced mathematics, 2012. doi: 10.1017/CBO9781139108133.
[6]	F. Morgan, Immiscible fluid clusters in $\mathbb R^2$ and $\mathbb R^3$, Michigan Math. J., 45 (1998), 441-450. doi: 10.1307/mmj/1030132292.
[7]	M. Ritoré and E. Vernadakis, Isoperimetric inequalities in Euclidean convex bodies, Trans. Amer. Math. Soc., 367 (2015), 4983-5014. doi: 10.1090/S0002-9947-2015-06197-2.
[8]	T. Schmidt, Strict interior approximation of sets of finite perimeter and functions of bounded variation, Proc. Am. Math. Soc., 143 (2015), 2069-2084. doi: 10.1090/S0002-9939-2014-12381-1.
[9]	A. Voß-Böhme and A. Deutsch, The cellular basis of cell sorting kinetics, J. Theoret. Biol., 263 (2010), 419-436. doi: 10.1016/j.jtbi.2009.12.011.
[10]	B. White, Existence of least-energy configurations of immiscible fluids, J. Geom. Analysis, 6 (1996), 151-161. doi: 10.1007/BF02921571.

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References:

[1]	L. Ambrosio and A. Braides, Functionals defined on partitions in sets of finite perimeter. II, J. Math. Pures Appl., 69 (1990), 307-333.
[2]	E. Bretin and S. Masnou, A new phase field model for inhomogeneous minimal partitions, and applications to droplets dynamics, Preprint, 2015.
[3]	S. Esedoglu and F. Otto, Threshold dynamics for networks with arbitrary surface tensions, Comm. Pure Appl. Math., 68 (2015), 808-864. doi: 10.1002/cpa.21527.
[4]	G. P. Leonardi, Infiltrations in immiscible fluids systems, Proc. Roy Soc. Edimburgh, 131 (2001), 425-436. doi: 10.1017/S0308210500000937.
[5]	F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems, Cambridge studies in advanced mathematics, 2012. doi: 10.1017/CBO9781139108133.
[6]	F. Morgan, Immiscible fluid clusters in $\mathbb R^2$ and $\mathbb R^3$, Michigan Math. J., 45 (1998), 441-450. doi: 10.1307/mmj/1030132292.
[7]	M. Ritoré and E. Vernadakis, Isoperimetric inequalities in Euclidean convex bodies, Trans. Amer. Math. Soc., 367 (2015), 4983-5014. doi: 10.1090/S0002-9947-2015-06197-2.
[8]	T. Schmidt, Strict interior approximation of sets of finite perimeter and functions of bounded variation, Proc. Am. Math. Soc., 143 (2015), 2069-2084. doi: 10.1090/S0002-9939-2014-12381-1.
[9]	A. Voß-Böhme and A. Deutsch, The cellular basis of cell sorting kinetics, J. Theoret. Biol., 263 (2010), 419-436. doi: 10.1016/j.jtbi.2009.12.011.
[10]	B. White, Existence of least-energy configurations of immiscible fluids, J. Geom. Analysis, 6 (1996), 151-161. doi: 10.1007/BF02921571.

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