Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I am studying some papers about the fractional laplacian, and I am stuck on a formula that I do not understand. I would like to ask if anybody can give me some help. In this paper, on page 12, there is a formula for the Green function $G_\alpha$ of a problem in $1+1$ dimensions. However, in this paper by Xavier Cabré, it seems to me that the Green function should have a completely different form, as noticed in Remarks 3.8 and 3.10. In particular, I do not understand the presence of an exponential term in the paper by Kenig. So my question is: what is the relationship between the function $G_\alpha$ of the first paper and the functions appearing in Remarks 3.8 and 3.10 of the second paper?

share|improve this question

1 Answer 1

It looks like Kenig-Martel-Robbiano study a parabolic problem while Cabré-Sire study an elliptic problem. For ordinary Laplacian, the heat kernel $t^{-n/2}e^{-|x|^2/(4t)}$ does not look very much like the fundamental solution $|x|^{2-n}$ either.

share|improve this answer
    
After some more thoughts, I think they solve different problems. While Cabré solves $(-\Delta)^su=h$, Kenig et al. solve $(-\Delta)^s u+u=h$. It is not trivial to move from one equation to the other. Even for the ordinary laplacian, the Green function for $-\Delta+I$ decays exponentially fast at infinity, while the Green function for $-\Delta$ decays like a negative power. But there is still something I do not understand completely. –  Siminore Aug 18 '12 at 8:04

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.