On the $p$-harmonic Robin radius in the Euclidean space

For $p>1$, the notion of the $p$-harmonic Robin radius is introduced in the space $\mathbb R^n$, $n\geq2$. If the corresponding part of the boundary degenerates the Robin–Neumann radius is considered. The monotonicity of the $p$-harmonic Robin radius under some deformations of a domain is proved. In the Euclidean space, some extremal decomposition problems are solved. The definitions and proofs are based on the technique of modules of curve families.