Maximum principles, Harnack inequality for classical solutions

where the matrix (aij) is symmetric, and uniformly elliptic (aij) ≥ γI, for some γ > 0. In terms of regularity of the coefficients, let us assume the are continuous functions. As opposed to the interior and boundary regularity estimates, where we worked with integral quantities, here we work with pointwise estimates on the solution. The point is the following: assume u ∈ C2(Ω) attains a maximum at x0 ∈ Ω. Then we can immediately conclude that the gradient of u vanishes at x0, and the Hessian of u is non-positive definite at x0, i.e.