Two geometric lemmas for S N - 1 -valued maps and an application to the homogenization of spin systems
ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 11

We prove two geometric lemmas for 𝒮$$-valued functions that allow to modify sequences of lattice spin functions on a small percentage of nodes during a discrete-to-continuum process so as to have a fixed average. This is used to simplify known formulas for the homogenization of spin systems.

DOI : 10.1051/cocv/2021007
Classification : 35B27, 74Q05, 49J45
Keywords: Spin systems, maps with values on the sphere, lattice energies, discrete-to-continuum, homogenization 
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     author = {Braides, Andrea and Vallocchia, Valerio},
     editor = {Buttazzo, G. and Casas, E. and de Teresa, L. and Glowinski, R. and Leugering, G. and Tr\'elat, E. and Zhang, X.},
     title = {Two geometric lemmas for $S^{N - 1}$-valued maps and an application to the homogenization of spin systems},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     year = {2021},
     publisher = {EDP-Sciences},
     volume = {27},
     doi = {10.1051/cocv/2021007},
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Braides, Andrea; Vallocchia, Valerio. Two geometric lemmas for $S^{N - 1}$-valued maps and an application to the homogenization of spin systems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 11. doi: 10.1051/cocv/2021007

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Cité par Sources :

The authors acknowledge the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.

Dedicated to Enrique Zuazua on the occasion of his 60th birthday.