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


35F20 Nonlinear first-order PDEs
Full Text: DOI Numdam EuDML


[1] Barthelemy, L.), Benilan, Ph.).- Sous-potentiels non linéaires. (à paraître).
[2] Benilan, Ph.).- Equations d’évolution dans un espace de Banach et applications. Thèse d’Etat, Université d’Orsay, 1972.
[3] Benilan, Ph.), Crandall, M.G.), Pazy, A.).- Evolution equation governed by accretive operators. (livre en préparation).
[4] Diaz, J.I.), Veron, L.). - Existence theory and qualitive properties of solutions of some first order quasilinear variational inequalities, Indiana Un. Math. J., t. 32, N°3, 1983. · Zbl 0488.35042
[5] Kruskov, S.N.).- First order quasilinear equation in several independent variables, Math. U.R.S.S. Sb, t. 10, 1970, p. 217-243. · Zbl 0215.16203
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.