Conormal flux of a divergence-form operator (${\sum }_{i,j}{a}_{ij}{D}_{i}u{\nu }_j$)

For sufficiently regular coefficients and a function $u$, this boundary quantity is the outward flux appearing when a divergence-form principal operator is integrated by parts. Its zero condition is the natural Neumann boundary condition for the corresponding bilinear form. With only bounded coefficients and a Sobolev solution, the weak formulation can encode that natural condition without asserting that a pointwise boundary derivative exists.