Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Continuous harmonic functions on a ball that are not in $H^s$ for
  $s>1/2$   We show that there are harmonic functions on a ball ${\mathbb{B}_n}$ of
$\mathbb{R}^n$, $n\ge 2$, that are continuous up to the boundary (and even
H\"older continuous) but not in the Sobolev space $H^s(\mathbb{B}_n)$ 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 $n\ge 2$ 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.

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The categories of this paper are: adap-org
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The categories of this paper are: cmp-lg
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The categories of this paper are: econ
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The categories of this paper are: solv-int
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The categories of this paper are: q-fin
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The categories of this paper are: atom-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: stat
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The categories of this paper are: physics
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The categories of this paper are: funct-an
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The categories of this paper are: q-alg
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The categories of this paper are: chem-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: plasm-ph
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The categories of this paper are: math-ph
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The categories of this paper are: cs