# Harnack Inequality

In the Han and Lin book there is a Harnak inequality for elliptic operators of the kind: $$L u = D_i \big(a^{ij}\, D_ju\big),$$ and the constant $C$ in the Harnack inequality does not depend on the radius of the ball on which I take the $\sup$ and the $\inf$. (Theorem 4.17, Elliptic Partial Differential Equations, Han & Lin)

In the book of Gilbarg and Trudinger, there is another version of the Harnkack inequality for elliptic operators of the kind: $$L u = D_i\big(a^{ij} \,D_j u + b^j \,u \big) + c^i\,D_iu + d\, u$$ but in this case the constant does depend on the size of the ball (Theorem 8.20, Elliptic Partial Differential Equations of Second Order, Gilbarg & Trudinger).

I need something in between. Namely I need an Harnack inequality for operator of the kind $$L u = D_i\big(a^{ij} \,D_j u \Big) + c^i \, D_iu$$ but such that the constant does not depend on the radius of the ball. Does exist such result?

Thanks!

## 1 Answer

Ok, I agree with you. Here is the link http://www.sciencedirect.com/science/article/pii/S0022039608001770

• It would help if you included the desired inequality in your answer, links may become invalid over time. Aug 29, 2017 at 14:30