Consider a Brownian motion starting at an interior point of the maximum or minimum domains with regular varying boundary, namely, D-max = {( x, y(1), y(2)) : parallel to x parallel to < max(i=1,2){f(i)(1 + y(i))}} and D-min = {(x, y(1), y(2)) : parallel to x parallel to < min(i=1,2){f(i) (1 + y(i))}}, in Rd+2, d >= 1, respectively, where parallel to.parallel to is the Euclidean norm in R-d, y(1), y(2) >= -1, and f(i) are regularly varying at infinity. Let tau(Dmax) and tau(Dmin) denote the first times the Brownian motion exits from D-max and D-min. Estimates with exact constants for the asymptotics of log P(tau(Dmax) > t) and log P(tau(Dmin) > t) are given as t -> infinity, depending on the relationship between f(1) and f(2), respectively. The proof methods are based on Gordon's inequality and early works of Li, Lifshits, and Shi in the single general domain case.
