New thoughts on Weinberger’s first and second integral bounds for Green’s function

Abstract

New thoughts about the first and second integral bounds of Hans F. Weinberger for Green’s functions of uniformly elliptic equations are presented by extending the bounds to two optimal monotone principles, but also further explored via: (i) discovering two new sharp Green-function- involved isoperimetric inequalities; (ii) verifying the lower dimensional Polya 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.