  We introduce and analyse the notion of slice continuity between operators on
Banach spaces in the setting of the Daugavet property.
  It is shown that under the slice continuity assumption the Daugavet equation
holds for weakly compact operators. As an application we define and
characterise the Daugavet property for bilinear maps, and we prove that this
allows us to describe some $p$-convexifications of the Daugavet equation for
operators on Banach function spaces that have recently been introduced.
