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This paper is concerned with the incompressible Euler equations. In Onsager'scritical classes we provide explicit formulas for the Duchon-Robert measure interms 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 $L^\infty_{x,t}\cap L^1_tBV_x$ solutions, in both the absence or presence of a physical boundary. Thisresult generalises the previously known case of Vortex Sheets, showing thatenergy conservation follows from the structure of $L^\infty\cap BV$incompressible vector fields rather than the flow having "organizedsingularities". The interior energy conservation features the use of Ambrosio'sanisotropic optimization of the convolution kernel and it differs from theusual energy conservation arguments by heavily relying on the incompressibilityof the vector field. This is the first energy conservation proof, for a givenclass of solutions, which fails to simultaneously apply to both compressibleand incompressible models, coherently with compressible shocks havingnon-trivial entropy production. To run the boundary analysis we introduce anotion of "normal Lebesgue trace" for general vector fields, very reminiscentof the one for $BV$ functions. We show that having such a null normal trace isbasically equivalent to have vanishing boundary energy flux. This goes beyondthe previous approaches, laying down a setup which applies to every Lipschitzbounded domain. Allowing any Lipschitz boundary introduces severaltechnicalities to the proof, with a quite geometrical/measure-theoreticalflavour.

Dissipation in Onsager's critical classes and energy conservation in $BV\cap L^\infty$ with and without boundary