Calculus of Variations and Geometric Measure Theory

A. Braides - M. Cicalese - F. Solombrino

Q-tensor continuum energies as limits of head-to-tail symmetric spins systems

created by solombrin on 15 Oct 2013
modified on 06 Aug 2015


Published Paper

Inserted: 15 oct 2013
Last Updated: 6 aug 2015

Journal: SIAM J. Math. Anal.
Volume: 47
Number: 4
Pages: 2832-2867
Year: 2015
Doi: 10.1137/130941341


We consider a class of spin-type discrete systems and analyze their continuum limit as the lattice spacing goes to zero. Under standard coerciveness and growth assumptions together with an additional head-to-tail symmetry condition, we observe that this limit can be conveniently written as a functional in the space of $Q$-tensors. We further characterize the limit energy density in several cases (both in $2$ and $3$ dimensions). In the planar case we also develop a second-order theory and we derive gradient or concentration-type models according to the chosen scaling.