PDE comparison principles for Robin problems

Abstract

We compare the solutions of two Poisson problems in a spherical shell with Robin boundary conditions, one with given data, and one where the data has been cap symmetrized. When the Robin parameters are nonnegative, we show that the solution to the symmetrized problem has larger convex means. Sending one of the Robin parameters to +∞, we obtain mixed Robin/Dirichlet comparison results in shells. We prove similar results on balls and prove a comparison principle on generalized cylinders with mixed Robin/Neumann boundary conditions.

Extremizing temperature functions of rods with Robin boundary conditions

“…In addition to the the results of [1,2,3], in [17] the first author studies Poisson problems of the form…”

“…To the best of our knowledge, references [1,2,3,17] comprise all that has appeared in print to addresses Robin comparison principles for differential equations in the spirit of Talenti. Thus, our work adds an interesting contribution to this new direction in the study of comparison principles.…”

Extremizing Temperature Functions of Rods with Robin Boundary Conditions

“…In addition to the the results of [1], [2], and [3], in [16] the first author studies Poisson problems of the form…”

“…To the best of our knowledge, references [1], [2], [3], and [16] comprise all that has appeared in print to addresses Robin comparison principles for differential equations in the spirit of Talenti. Thus, our work adds an interesting contribution to this new direction in the study of comparison principles.…”