Symmetry of a boundary integral operator and a characterization of a ball. (English) Zbl 1003.31001

Let \[ K_\Omega f(x)= \frac{1}{\omega_n}\int_{\partial \Omega}\frac{\langle y-x,\nu(y)\rangle}{|y-x|^{n}}f(y) d\sigma(y), \qquad x\in \partial \Omega \] be the singular integral operator generated by the double layer potential of \(f\in L_2(\partial \Omega)\), where \(\Omega\) is a bounded domain in \(\mathbb{R}^n\). In the case where \(\Omega\) is \(C^1\), it is known that the operator \(K_\Omega\) may be self-adjoint only in the case of a ball, which follows from the fact that the condition \[ \langle y-x,\nu(y)\rangle=\langle x-y,\nu(y)\rangle \] providing the selfadjointness of the operator \(K_\Omega\) is valid for balls only [H. P. Boas, Math. Ann. 248, 275-278 (1980; Zbl 0414.32002)]. The author shows that the same is valid in the case of Lipschitz domains. For such domains he also proves the statement \[ K_\Omega f\equiv 0 \quad \forall f\in L_2(\partial \Omega) \quad \text{with mean value zero} \quad \Longrightarrow \quad n=2 \;\text{and} \Omega \;\text{is a disc}. \]


31B20 Boundary value and inverse problems for harmonic functions in higher dimensions
47G10 Integral operators


Zbl 0414.32002