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Fractional calculus for power functions and eigenvalues of the fractional Laplacian. (English) Zbl 1312.35176
Summary: We calculate the fractional Laplacian \(\Delta^{\alpha/2}\) for functions of the form \(u(x) = (1 - |x|^2)_+^p\) and \(v(x) = x_d u(x)\). As an application, we estimate the first eigenvalues of the fractional Laplacian in a ball.

MSC:
35R11 Fractional partial differential equations
35P15 Estimates of eigenvalues in context of PDEs
31C25 Dirichlet forms
35B10 Periodic solutions to PDEs
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