An extension problem related to the fractional Laplacian. (English) Zbl 1143.26002

The authors relate the fractional Laplacian of a function \(f: \mathbb{R}^{n}\rightarrow \mathbb{R}\) to solutions \(u:\mathbb{R}^{n}\times [0,\infty )\rightarrow \mathbb{R}\) of the extension problem \[ \left\{ \begin{matrix} u(x,0)=f(x) \\ \Delta _{x}u+\frac{a}{y}u_{y}+u_{yy}=0. \end{matrix} \right. \] It is shown that \[ \lim_{y\rightarrow 0}y^{a}u_{y}(x,y)=u_{z}(x,0)=-(-\Delta )^{s}f(x) \] where \(s=\frac{1-a}{2}\) and \(z=\left( \frac{y}{1-a}\right) ^{1-a}.\) This work extends the well-known fact that the operator \((-\Delta )^{1/2}\) can be obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to the Neumann condition. Therefore, the present work generalizes this characterization to general fractional powers of the Laplacian. This is also done for other integro-differential operators and some properties of these integro-differential equations are derived.


26A33 Fractional derivatives and integrals
35J70 Degenerate elliptic equations
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