Calculus of Variations and Geometric Measure Theory

L. Ambrosio - C. De Lellis - J. MalĂ˝

On the chain rule for the divergence of BV like vector fields: applications, partial results, open problems

created by ambrosio on 10 Jan 2005
modified by delellis on 02 Jul 2013


Published Paper

Inserted: 10 jan 2005
Last Updated: 2 jul 2013

Journal: Perspective in nonlinear partial differential equations
Pages: 31-67
Year: 2007

To appear in the forthcoming book by the AMS series in contemporary mathematics ``Perspectives in Nonlinear Partial Differential Equations: in honor of Haim Brezis''


We discuss the problem of computing the distributional divergence of a vector field of the form h(w)B, where $h$ is a smooth scalar function and B is a BV vector field, knowing the distributional divergence of all vector fields wiB, where wi are the components of w.

We present partial results on this problem, conjectures, and links with other problems related to the SBV regularity of solutions of Hamilton-Jacobi equations and systems of conservation laws, and a conjecture recently made by Bressan.

For the most updated version and eventual errata see the page


Keywords: BV functions, Chain Rule, Renormalized solutions, continuity equation