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Uniqueness of $$L^\infty$$ solutions for a class of conormal $$BV$$ vector fields. (English) Zbl 1064.35033
Chanillo, Sagun (ed.) et al., Geometric analysis of PDE and several complex variables. Dedicated to François Trèves. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3386-3/pbk). Contemporary Mathematics 368, 133-156 (2005).
Summary: Let $$X$$ be a bounded vector field with bounded divergence defined in an open set $$\Omega$$ of $$\mathbb{R}^d$$, transverse to a hypersurface $$S$$. Let $$\Omega_0$$ be an open subset of $$\Omega$$ such that the Hausdorff measure $${\mathcal H}^{d-1} (\Omega\setminus \Omega_0)=0$$. The authors suppose that the vector field $$X$$ belongs to $$BV_{\text{loc}}(\Omega_0)$$ “conormally”, an assumption made precise in the text, which is satisfied whenever the gradients of the coefficients of $$X$$ have locally only a single component which is actually a Radon measure. This class can be invariantly defined and contains the so-called piecewise $$W^{1,1}$$ functions studied in P.-L. Lions [C. R. Acad. Sci., Paris, Sér. I, Math. 326, 833–838 (1998; Zbl 0919.34028)]. They prove the uniqueness of $$L^\infty$$ solutions for the Cauchy problem related to $$X$$ across the hypersurface $$S$$. They use for the proof some simple arguments of geometric measure theory to get rid of closed sets with codimension $$>1$$. Next, we need an anisotropic regularization argument analogous to the one used in F. Bouchut [Arch. Ration. Mech. Anal. 157, 75–90 (2001; Zbl 0979.35032)].
For the entire collection see [Zbl 1058.35003].

##### MSC:
 35F05 Linear first-order PDEs 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations 26A45 Functions of bounded variation, generalizations
##### Keywords:
Cauchy problem; anisotropic regularization argument