  In this paper, positive solutions to the Laplace equation with 1-dimensional
circular singularities are investigated. First, we establish $L^p$
integrability estimates for such solutions $u$ near the singularities, in
comparison with classical $L^1$ estimates. Then we characterize $-\Delta u$
around the circle. The method we developed may also be adapted to higher
dimensional as well as more generalized cases.
