  We propose new entropy admissibility conditions for multidimensional
hyperbolic scalar conservation laws with discontinuous flux which generalize
one-dimensional Karlsen-Risebro-Towers entropy conditions.
  These new conditions are designed, in particular, in order to characterize
the limit of vanishing viscosity approximations.
  On the one hand, they comply quite naturally with a certain class of physical
and numerical modeling assumptions; on the other hand, their mathematical
assessment turns out to be intricate. \smallskip The generalization we propose
is not only with respect to the space dimension, but mainly in the sense that
the "crossing condition" of [K.H. Karlsen, N.H. Risebro, J. Towers,
Skr.\,K.\,Nor.\,Vid.\,Selsk. (2003)] is not mandatory for proving uniqueness
with the new definition. We prove uniqueness of solutions and give tools to
justify their existence via the vanishing viscosity method, for the
multi-dimensional spatially inhomogeneous case with a finite number of
Lipschitz regular hypersurfaces of discontinuity for the flux function.
