Wednesday, March 19, 2025 - 14:40 in V2-205
Isolated singularities: Bocher type theorem for elliptic equations with drift perturbed Levy operator
A talk in the Nonlocal Equations: Analysis and Numerics series by
Tomasz Klimsiak from Torun
| Abstract: |
A classical Bocher’s theorem asserts that any positive harmonic function
(with respect to the Laplacian) in the unit punctured ball can be expressed,
up to a multiplication constant, as the sum of the Newtonian kernel and
a positive function that is harmonic in the whole unit ball. This theorem
expresses one of the fundamental results in the theory of isolated singularities and it can be viewed as a statement on the asymptotic behavior of
positive harmonic functions near their isolated singularities. We generalize
this results to drift perturbed Levy operators. We propose a new approach
based on the probabilistic potential theory. It applies to Levy operators for
which the resolvent of its perturbation is strongly Feller. In particular our
result encompasses drift perturbed fractional Laplacians with any stability
index bounded between zero and two - the method therefore applies to
subcritical and supercritical cases.
Within the CRC this talk is associated to the project(s): A7 |
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