A GENERAL HARNACK INEQUALITY FOR SEMILINEAR SUBELLIPTIC EQUATIONS

In this talk, I will present a generalized Harnack inequality for non-negative weak solutions of divergence-form subelliptic equations on Carnot groups of the form, −X^∗(A(x)Xu)= f (u), where A(x) is uniformly elliptic and f : [0, ∞) →[0, ∞) is continuous and non-decreasing. The proof combines energy estimates for sublevel sets, the Gauss–Green theorem for divergence-measure horizontal fields, and the coarea formula in Carnot groups. Under an additional growth assumption on f , this approach yields the classical Harnack inequality. This is joint work with Ahmed Mohammed (Ball State University).