10 - Comparison Principles for Semilinear Poisson PDEs

Summary

Chapter 10 establishes comparison principles for solutions of partial differential equations. The prototypical result says that the solution of Poisson's equation gets bigger in an integral sense when the data in the equation is rearranged. Such comparisons have been used in the literature for deriving sharp bounds on certain eigenvalues, obtaining a priori bounds on solutions, and comparing Green functions, among other uses. These integral norm comparisons follow from star function comparisons, and so the task is to prove that rearranging the data in Poisson's equation increases the star function of the solution. The key is a maximum principle argument applied to the difference of star functions, making use of subharmonicity results from the preceding chapter.