We study an energy minimization problem ₈ ₉ W (zᵢ - zⱼ) for N points \z₁, , zN\ with applications in dislocation theory. The N points lie in the two-dimensional domain R -π, π, %who are trying to minimize their interaction energy where where the kernel W is derived from the Volterra potential V (x, y) = x²x²+y²-12 (x²+y²). We prove that the minimum energy is given by - N N +O (N). This lower bound recovers the leading order term of the Read-Shockley law characterizing the energy of small angle grain boundaries in polycrystals.
Grabner et al. (Mon,) studied this question.