when consider the fractional laplacian $(-\triangle)^\alpha$,is there an essential difference between $0<\alpha<1$ and $\alpha>1$ ? As far as I'm concern,the higer order laplacian ($\alpha>1$ ) ,unlike the lower case, has little connection with stochastic process.(lack of positivy.) Since most paper i have met is the case of $0<\alpha<1$.And i wonder how things change when considering the higer order laplacian?
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3 Answers
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The fractional Laplacian can be represented as a hypersingular integral operator, an integral operator with a singularity higher than the space dimension. Its convergence is provided by a regularization whose form depends on the order. From its form the difference in positivity properties is immediately clear. See S. G. Samko, Hypersingular Integrals and their applications, Taylor and Francis, London, 2002.
On the other hand, there exists a theory of parabolic equations with fractional Laplacians (and more general pseudodifferential operators) of any order. See S. D. Eidelman, S. D. Ivasyshen, and A. N. Kochubei, Analytic methods in the theory of differential and pseudo-differential equations of parabolic type, Birkhauser, Basel, 2004.
answered Apr 20, 2012 at 16:30
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$\begingroup$ thanks for the useful reference.i'm considering the fractional shrodinger equation and it's $L^p$ estimate ,so i first want to know more about it'e kernel $\endgroup$ Apr 20, 2012 at 16:45
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$\begingroup$ i try to look for that book,but it's not available.. I'm really appreciated that if you can send me a copy of the pdf file of your book mentioned above .here is my adress: [email protected]. thank you very much! $\endgroup$ Apr 24, 2012 at 15:14
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Well, the answer is in your question. When $0<\alpha<1$, and only then, $(-\Delta)^\alpha$ has a positive inverse. In particular, equations like $$(-\Delta)^\alpha u=f$$ or $$\partial_tu+(-\Delta)^\alpha u=0$$ satisfy a maximum principle. You get immediately an $L^p$ estimate for every $1\le p\le\infty$. If the equation is quasi-linear instead, it is likely that you have a $BV$-estimate for free. This helps a lot in proving well-posedness of the Boundary-value problem or the Cauchy problem.
answered Apr 20, 2012 at 16:23
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I discovered a two-sided fractional derivative similar to the Grunwald-Letnikov derivative and defined for orders greater than -1. If the absolute value of the order is less than one, there is an inverse given by the same formulae. I proved that this derivative is equal to the Riesz potential, but does not have its problems. You can see the papers:
Ortigueira, M. D., “Riesz Potentials and Inverses via Centred Derivatives” International Journal of Mathematics and Mathematical Sciences, Volume 2006 (2006), Article ID 48391, pages 1-12. Ortigueira, M. D., “Fractional Central Differences and Derivatives”, Journal of Vibration and Control , 14(9–10): 1255–1266, 2008.
Prof. I. Podlubny made an implementation for this derivative that is in the site of Matlab. If you need you can contact me [email protected]
