MATHEMATICS

Some properties of a class of vector potentials with a singular kernel

E. H. Khalilov, V. O. Safarova

Azerbaijan State Oil and Industry University, Baku, Azerbaijan

Abstract: The counterexample constructed by A.M. Lyapunov shows that for potentials of a simple and double layer with continuous density, the derivative, generally speaking, does not exist. Therefore, the operators
$$ (A\lambda)(x)=-2\int_\Omega[n(x),[n(x), rot_x\{\Phi_k(x,y)\lambda(y)n(y)\}]]d\Omega_y, \quad x\in\Omega, $$
and
$$ (B\mu)(x)=2\int_\Omega[n(x), grad_x\{\Phi_k(x,y)\mu(y)\}]d\Omega_y, \quad x\in\Omega, $$
are not defined in the space of continuous functions, where $\Omega\subset R^3$ is the Lyapunov surface, $n(x)$ is the external unit normal at point $x\in\Omega$, and $\Phi_k(x,y)$ is the fundamental solution of the Helmholtz equation. The paper proves that if functions $\lambda(x)$ and $\mu(x)$ satisfy the Dini condition, then integrals $(A\lambda)(x)$ and $(B\mu)(x)$ exist in the sense of the Cauchy principal value. In addition, the validity of the A. Zygmund type estimate for the integrals $(A\lambda)(x)$ and $(B\mu)(x)$ is shown, and the boundedness of operators $A$ and $B$ in generalized Hölder spaces is proved.

Keywords: electrical boundary value problem, magnetic boundary value problem, vector potentials, Helmholtz equation, generalized Hölder space.

UDC: 517.2, 519.64

Received: 08.02.2025
Accepted: September 6, 2025

DOI: 10.17223/19988621/97/3