On some second order and fourth order elliptic overdetermined problems (Q1891374)

Some overdetermined boundary value problems for the harmonic equation \(\Delta u = 0\) in a tubular region \(\Omega_0 \backslash \overline \Omega_1\) \((\Omega_1 \subset \Omega_0)\) in \(\mathbb{R}^3\) are considered. Under appropriate assumptions the authors conclude that the only possible configuration of \(\Omega_0\) and \(\Omega_1\) is two coaxial circular cylinders. One overdetermined boundary value problem for a fourth order equation \(\Delta^2u = 1\) in \(\Omega \subset \mathbb{R}^n\) with following conditions on \(\partial\Omega : u = \Delta u = 0\), \(\partial u/ \partial n = C = \text{const}\) \((n\) denotes an exterior normal) is also investigated. It is proved that this boundary value problem is solvable only if \(\Omega\) is an \(n\)-ball.