Archive for Rational Mechanics and Analysis

, Volume 175, Issue 2, pp 245-267

First online:

Divergence-Measure Fields, Sets of Finite Perimeter, and Conservation Laws

  • Gui-Qiang ChenAffiliated withDepartment of Mathematics, Northwestern University Email author 
  • , Monica TorresAffiliated withDepartment of Mathematics, Northwestern University

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Divergence-measure fields in L 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 . 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 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.