Hello Santu,

Thanks for asking. The use of Hopf’s lemma turns out to be standard for PDErs :).

You may argue in a “geometric way” as follows: Suppose there does not exist such a constant ; hence the function decays faster than the distance function to the boundary. By decay I mean there exists a sequence of points converging to the boundary. Therefore, if you compare the distance to the boundary and the amplitude of , you will get a curve which is convex. Therefore, the directional derivative with respect to the outer normal at the limit point (on the boundary) tends to be non-negative, which contradicts to what Hopf’s lemma says.

For the second part of you question, this is standard too. Let me know if you still cannot figure out how.

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Hi, thanks for your question. It looks like we have a typo here since does not belong to any Sobolev space. I think we can change the definition of a little bit as follows

where is a cut-off function which equals in and outside the ball . Then the following estimate remains valid as shown below

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