The family of level sets of a harmonic function
Tóm tắt
Families of hypersurfaces that are level-set families of harmonic functions free of critical points are characterized by a local differential-geometric condition. Harmonic functions with a specified level-set family are constructed from geometric data. As a by-product, it is shown that the evolution of the gradient of a harmonic function along the gradient flow is determined by the mean curvature of the level sets that the flow intersects.
Tài liệu tham khảo
Flatto, L., D.J. Newman, and H.S. Shapiro. 1966. The Level Curves of Harmonic Functions. Transactions of the American Mathematical Society 123: 425–436.
Jerrard, R.P., and L.A. Rubel. 1963. On the Curvature of the Level Lines of a Harmonic Function. Proceedings of the American Mathematical Society 14: 29–32.