Green’s second identity for generalized Laplacians
Walter Rudin · Proceedings of the American Mathematical Society · 1951
Let J(P, r) denote the closed circular disc bounded by the circle C(P, r) with center at P, and radius r, in the plane.We define the generalized Laplacian of the function F at P by AP(P) = lim 4[m(F; P, r) -F(P)]/r2, r->0 where m(F; P, r) is the mean value of F on C(P, r).The upper and lower Laplacians A*F(P) and A*P(P) are defined likewise, with lim sup and lim inf in place of lim [3].1 If/G¿ in a bounded domain P, we define [3, p. 281] V»f(P) = --(f f(Q)g(P, Q)dQ (P in R), ¿V J J R