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September 2, 2022

Continuous harmonic functions on a ball that are not in Hs for s>1/2. SC 1431

We show that there are harmonic functions on a ball 𝔹n of ℝn, n ≥ 2, that are continuous up to the boundary (and even Hölder continuous) but not in the Sobolev space Hs(𝔹n) for any s sufficiently big. The idea for the construction of these functions is inspired by the two-dimensional example of a harmonic continuous function with infinite energy presented by Hadamard in 1906. To obtain examples in any dimension n ≥ 2, we exploit certain series of spherical harmonics. As an application, we verify that the regularity of the solutions that was proven for a class of boundary value problems with nonlinear transmission conditions is, in a sense, optimal.