  This paper is concerned with the incompressible Euler equations. In Onsager's
critical classes we provide explicit formulas for the Duchon-Robert measure in
terms of the regularization kernel and a family of vector-valued measures
$\{\mu_z\}_z$, having some H\"older regularity with respect to the direction
$z\in B_1$. Then, we prove energy conservation for $L^\infty_{x,t}\cap L^1_t
BV_x$ solutions, in both the absence or presence of a physical boundary. This
result generalises the previously known case of Vortex Sheets, showing that
energy conservation follows from the structure of $L^\infty\cap BV$
incompressible vector fields rather than the flow having "organized
singularities". The interior energy conservation features the use of Ambrosio's
anisotropic optimization of the convolution kernel and it differs from the
usual energy conservation arguments by heavily relying on the incompressibility
of the vector field. This is the first energy conservation proof, for a given
class of solutions, which fails to simultaneously apply to both compressible
and incompressible models, coherently with compressible shocks having
non-trivial entropy production. To run the boundary analysis we introduce a
notion of "normal Lebesgue trace" for general vector fields, very reminiscent
of the one for $BV$ functions. We show that having such a null normal trace is
basically equivalent to have vanishing boundary energy flux. This goes beyond
the previous approaches, laying down a setup which applies to every Lipschitz
bounded domain. Allowing any Lipschitz boundary introduces several
technicalities to the proof, with a quite geometrical/measure-theoretical
flavour.
