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Bi- subharmonic distributions on \(\mathbb{R}^ n\) (\(n\geq 2\)). (Distributions bi-sousharmoniques sur \(\mathbb{R}^ n\) (\(n\geq 2\)).) (French) Zbl 0802.31001
A locally summable function \(\omega\) on \(\mathbb{R}^ n\), \(n\geq 2\), is said to be bi-subharmonic if \(\Delta^ 2 \omega\geq 0\), where \(\Delta\) is the Laplacian as a distribution. The author proves that a bi-subharmonic function is representable as a difference of two functions that are simultaneously bi-subharmonic and subharmonic if and only if \(\Delta\omega\) coincides almost everywhere with a subharmonic function admitting a superharmonic minorant that is bounded above. There are also characterizations of biharmonic functions \(\omega\) on \(\mathbb{R}^ 3\) (\(\Delta^ 2 \omega=0\)) in terms of their averages over balls or spheres.
MSC:
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
31B30 Biharmonic and polyharmonic equations and functions in higher dimensions
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