New thoughts on Weinberger's first and second integral bounds for Green's functions (Q1932244)

In the paper under review, the author proposes new views on the first and second integral bounds of Hans F. Weinberger for the Green functions of uniformly elliptic operators by extending the bounds to two optimal monotone principles. In particular, applications of the ideas developed are given through: (i) discovering two new sharp Green-function involved isoperimetric inequalities; (ii) verifying the lower dimensional Pólya conjecture for the lowest eigenvalue of the Laplacian; (iii) sharpening an eccentricity-based lower bound for the Mahler volumes of the origin-symmetric convex bodies.