Harnack inequality and maximum principle for degenerate Kolmogorov operators in divergence form

Abstract

We prove the parabolic strong maximum principle and the Harnack inequality for classical solutions to degenerate second order partial differential equations of the formLu:=∑i,j=1m0∂xi(aij∂xju)+∑j=1m0bj∂xju+cu+∑i,j=1Nbijxj∂xiu−∂tu=f, where 1≤m0≤N, A0=(aij)i,j=1,…,m0 is a bounded, symmetric and uniformly positive matrix and the matrix B:=(bij)i,j=1,…,N has real constant entries. Moreover, we assume low regularity (i.e. Hölder continuity) on the coefficients aij, bj and c, for i,j=1,…,m0. We point out the proofs of our main results only rely on the classical theory developed for harmonic functions.