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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!

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

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  • $\begingroup$ It would help if you included the desired inequality in your answer, links may become invalid over time. $\endgroup$ – Alex M. Aug 29 '17 at 14:30

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