Divergence-measure fields, sets of finite perimeter, and conservation laws

Divergence-measure fields in L-infinity over sets of finite perimeter are analyzed. A notion of normal traces over boundaries of sets of finite perimeter is introduced, and the Gauss-Green formula over sets of finite perimeter is established for divergence-measure fields in L-infinity. The normal trace introduced here over a class of surfaces of finite perimeter is shown to be the weak-star limit of the normal traces introduced in CHEN & FRID [6] over the Lipschitz deformation surfaces, which implies their consistency. As a corollary, an extension theorem of divergence- measure fields in L-infinity over sets of finite perimeter is also established. Then we apply the theory to the initial-boundary value problem of nonlinear hyperbolic conservation laws over sets of finite perimeter.