Jeffrey J. Langford, Assistant Professor of Mathematics
In this paper we prove a spherical comparison result for the (k,n)-spherical rearrangement using the spherical Green’s function and a rearrangement inequality of A. Baernstein. We next use a simple reflection argument to obtain a Neumann comparison result on a hemisphere for the (k,n)-hemispherical rearrangement. Using the Lambert equal-area projection and stereographic projection, we obtain Neumann comparison results in Euclidean space. We end with a discussion of open problems in the unit disk involving heat flow.
Langford, Jeffrey J. “Neumann Comparison Theorems in Elliptic PDEs.” Potential Analysis 43, no. 3 (2015) : 415-459.