Characterizing the strong maximum principle for constant coefficient subequations

In this paper we characterize the degenerate elliptic equations F(D ² u) = 0 whose subsolutions (F(D ² u) ≥ 0) satisfy the strong maximum principle. We introduce an easily computed function f on (0,1) which is determined by F, and we show that the strong maximum principle holds depending on whether ^R _{0+ f} ^dy ₍ y ₎ is infinite or finite. This is in the spirit of previous work characterizing the ordinary maximum principle in terms of the geometry of the set of symmetric matrices F = {F ≥ 0}. Along the way, radial subsolutions are characterized, and, as an application, a sufficient condition for strong comparison is established. A number of examples, important for the theory of such equations, are examined.