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Matematikcentrum

Lunds universitet

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Fractional-order operators and integration by parts

There has recently been established a solvability theory for boundary value problems defined on a subset $\Omega$ of $R^n$, for the fractional Laplacian $(-\Delta)^a$, $0\lt a\lt 1$, and more generally for even elliptic pseudodifferential operators on $R^n$ of order $2a$. Here it is mostly the homogeneous Dirichlet problem that has been discussed, but it is also possible to define nonzero Dirichlet and Neumann boundary values. We shall present elements of the linear solvability theory, and then go on to the latest development: Integration by parts formulas, that are useful in the study of nonlinear problems.