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Dirichlet problems for the biharmonic equation. (English) Zbl 1136.31001

The author constructs an explicit solution of the biharmonic equation \[ (\partial_z \partial_{\overline z})^2 w = f \] in the unit disk \(\mathbb D =\{ z\in \mathbb C:\, | z|<1\}\) for two different sets of boundary conditions:
(i)\(w=\varphi_0\), \(\partial_z \partial_{\overline z} w =\varphi_1\) on \(\partial \mathbb D\);
(ii)\(w=\varphi_0\), \(\partial_{\overline z} w =\varphi_1\) on \(\partial \mathbb D\).

MSC:

31A30 Biharmonic, polyharmonic functions and equations, Poisson’s equation in two dimensions
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