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The obstacle problem for a first-order quasilinear equation. (Problème d’obstacle pour une équation quasi-linéaire du premier ordre.) (French) Zbl 0631.47038

Let \(\phi\) be a continuous map from \({\mathbb R}\) into \({\mathbb R}^ N\) and \(\psi\) be any measurable obstacle. We study the problem \[ \text{div}\, \phi (u)\leq f(.,u),\quad u\leq \psi, \] where a solution of this problem is understood as a subsolution in the Kruzhkov sense of \(\text{div}\,\phi (u)=f(.,u)\) on \({\mathbb R}^ N\). We show, under some assumptions on \(f\), that, if it has one, this problem has a largest solution. In the same way, we show existence of a largest solution for the associated evolution problem \[ \partial u/\partial t+ \text{div}\, \phi (u)\leq f(.,u), \quad u\leq \psi,\quad u(0,.)\leq u_ 0. \]
Reviewer: Louise Barthélemy

MSC:

35F20 Nonlinear first-order PDEs
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References:

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