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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].

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