Necessary condition for eigensolutions of the 1-Laplace operator by means of inner variations

Eigenfunctions of the $p$ -Laplace operator for $p>1$ are defined to be critical points of an associated variational problem or, equivalently, to be solutions of the corresponding Euler–Lagrange equation. In the highly degenerated limit case of the 1-Laplace operator eigenfunctions can also be defined to be critical points of the corresponding variational problem if critical points are understood on the basis of the weak slope. However, the associated Euler–Lagrange equation has many solutions that are not critical points and, thus, it cannot be used for an equivalent definition. The present paper provides a new necessary condition for eigenfunctions of the 1-Laplace operator by means of inner variations of the associated variational problem and it is shown that this condition rules out certain solutions of the Euler–Lagrange equation that are not eigenfunctions.