Calculus of Variations and Geometric Measure Theory

L. Ambrosio - C. De Lellis

A note on admissible solutions of 1d scalar conservation laws and 2d Hamilton-Jacobi equations

created on 25 Jun 2004
modified by delellis on 05 May 2011


Published Paper

Inserted: 25 jun 2004
Last Updated: 5 may 2011

Journal: J. Hyperbolic Diff. Equ.
Volume: 1
Number: 4
Pages: 813-826
Year: 2004


Let $\Omega\subset *R*^2$ be an open set and let $f\in C^2 (*R*)$ with $f''>0$. In this note we prove that entropy solutions of $D_t u + D_x f(u) =0$ belong to $SBV_{loc} (\Omega)$. As a corollary we prove the same property for gradients of viscosity solutions of planar Hamilton--Jacobi PDEs with uniformly convex hamiltonians.

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Keywords: Hamilton-Jacobi equations, Scalar conservation laws, SBV functions, Hopf-Lax formula