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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

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 & Heterogeneous Media, 2016, 11 (3) : 501-508. doi: 10.3934/nhm.2016006
References:
[1]

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[2]

E. Bretin and S. Masnou, A new phase field model for inhomogeneous minimal partitions, and applications to droplets dynamics,, Preprint, (2015).   Google Scholar

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S. Esedoglu and F. Otto, Threshold dynamics for networks with arbitrary surface tensions,, Comm. Pure Appl. Math., 68 (2015), 808.  doi: 10.1002/cpa.21527.  Google Scholar

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F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems,, Cambridge studies in advanced mathematics, (2012).  doi: 10.1017/CBO9781139108133.  Google Scholar

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F. Morgan, Immiscible fluid clusters in $\mathbb R^2$ and $\mathbb R^3$,, Michigan Math. J., 45 (1998), 441.  doi: 10.1307/mmj/1030132292.  Google Scholar

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M. Ritoré and E. Vernadakis, Isoperimetric inequalities in Euclidean convex bodies,, Trans. Amer. Math. Soc., 367 (2015), 4983.  doi: 10.1090/S0002-9947-2015-06197-2.  Google Scholar

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T. Schmidt, Strict interior approximation of sets of finite perimeter and functions of bounded variation,, Proc. Am. Math. Soc., 143 (2015), 2069.  doi: 10.1090/S0002-9939-2014-12381-1.  Google Scholar

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A. Voß-Böhme and A. Deutsch, The cellular basis of cell sorting kinetics,, J. Theoret. Biol., 263 (2010), 419.  doi: 10.1016/j.jtbi.2009.12.011.  Google Scholar

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B. White, Existence of least-energy configurations of immiscible fluids,, J. Geom. Analysis, 6 (1996), 151.  doi: 10.1007/BF02921571.  Google Scholar

show all references

References:
[1]

L. Ambrosio and A. Braides, Functionals defined on partitions in sets of finite perimeter. II,, J. Math. Pures Appl., 69 (1990), 307.   Google Scholar

[2]

E. Bretin and S. Masnou, A new phase field model for inhomogeneous minimal partitions, and applications to droplets dynamics,, Preprint, (2015).   Google Scholar

[3]

S. Esedoglu and F. Otto, Threshold dynamics for networks with arbitrary surface tensions,, Comm. Pure Appl. Math., 68 (2015), 808.  doi: 10.1002/cpa.21527.  Google Scholar

[4]

G. P. Leonardi, Infiltrations in immiscible fluids systems,, Proc. Roy Soc. Edimburgh, 131 (2001), 425.  doi: 10.1017/S0308210500000937.  Google Scholar

[5]

F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems,, Cambridge studies in advanced mathematics, (2012).  doi: 10.1017/CBO9781139108133.  Google Scholar

[6]

F. Morgan, Immiscible fluid clusters in $\mathbb R^2$ and $\mathbb R^3$,, Michigan Math. J., 45 (1998), 441.  doi: 10.1307/mmj/1030132292.  Google Scholar

[7]

M. Ritoré and E. Vernadakis, Isoperimetric inequalities in Euclidean convex bodies,, Trans. Amer. Math. Soc., 367 (2015), 4983.  doi: 10.1090/S0002-9947-2015-06197-2.  Google Scholar

[8]

T. Schmidt, Strict interior approximation of sets of finite perimeter and functions of bounded variation,, Proc. Am. Math. Soc., 143 (2015), 2069.  doi: 10.1090/S0002-9939-2014-12381-1.  Google Scholar

[9]

A. Voß-Böhme and A. Deutsch, The cellular basis of cell sorting kinetics,, J. Theoret. Biol., 263 (2010), 419.  doi: 10.1016/j.jtbi.2009.12.011.  Google Scholar

[10]

B. White, Existence of least-energy configurations of immiscible fluids,, J. Geom. Analysis, 6 (1996), 151.  doi: 10.1007/BF02921571.  Google Scholar

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