On the asymptotic behavior of the discrete minimal energy on rectifiable sets
Abstract:
We consider a generalization of the classical Thomson's problem of finding ground state configurations and minimal potential energy of N electrons on the sphere. In our considerations the potential of the repelling force is proportional to the reciprocal of the power s>0 of the distance and the particles are restricted to a rectifiable compact set A. When s is greater than the Hausdorff dimension of A, the methods of potential theory do not work. In this case, using tools from geometric measure theory, we obtain the main term in the asymptotics of the minimal energy of N particles on A and the limit distribution of optimal configurations. Joint work with Doug Hardin, Edward Saff (Vanderbilt University).
Applied Mathematics Seminar