Abstract
In this note we show that for f ∈ C((0,∞); R+) ∩ C1 ((0,∞)) with support in [0,∞), if a function u ∈ C1(R2) is such that support (u+) is compact and u(x) = ∫R2 f(u(y)) log 1/(|x-y|)dy ∀ x, then u is radial. This result is important for some free boundary problems in R2 or some axisymmetric ones in Rn.
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Tadie Radially Symmetric Functions as Fixed Points of some Logarithmic Operators. Potential Analysis 9, 83–89 (1998). https://doi.org/10.1023/A:1008606430233
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DOI: https://doi.org/10.1023/A:1008606430233