Maximum principles, extension problem and inversion for nonlocal one-sided equations
By F. J. Martín-Reyes, Pablo Raúl Stinga, José L. Torrea. 2015
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We study one-sided nonlocal equations of the form $$\int_{x_0}^\infty\frac{u(x)-u(x_0)}{(x-x_0)^{1+\alpha}} dx=f(x_0),$$ on the real line. Notice that to compute this nonlocal operator of order $0<\alpha<1$ at a point $x_0$ we need to know the values of $u(x)$ to the right of $x_0$, that is, for $x\geq x_0$. We show that the operator above corresponds to a fractional power of a one-sided first order derivative. Maximum principles and a characterization with an extension problem in the spirit of Caffarelli--Silvestre and Stinga--Torrea are proved. It is also shown that these fractional equations can be solved in the general setting of weighted one-sided spaces. In this regard we present suitable inversion results. Along the way we are able to unify and clarify several notions of fractional derivatives found in the literature.
Subfields: Applied Mathematics
View Paper: https://www.semanticscholar.org/paper/816d8ef962c301ff9a8a510535072ba44d6cf88d
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In Applied Mathematics
Citations in the corpus
- Weyl and Marchaud Derivatives: A Forgotten History (2017, background): “The approach described in [12,14], and then in [15], was also extended to the case of fractional operators in [16] and independently also, as very often it happens when the time is ripe, in [17].”