Quasi-linear elliptic and parabolic equations in \(L^ 1\) with nonlinear boundary conditions. (English) Zbl 0882.35048

Let \(\Omega\) be a bounded domain in \(\mathbb{R}^N\) with smooth boundary \(\partial\Omega\). Under classical assumptions on the vector valued function \(a\), the authors use variational methods to deduce the existence and uniqueness of solutions to the problem: \[ u-\text{div }a(x,Du)= f\quad\text{in }\Omega,\quad -\partial u/\partial\eta_a\in \beta(u)\quad\text{on }\partial\Omega, \] where \(\beta\) is a maximal monotone graph in \(\mathbb{R}\times\mathbb{R}\) with \(0\in\beta(0)\) and \(f\in L^1(\Omega)\). They introduce completely accretive operators and characterize the closure of the smaller one by introducing the notion of entropy solutions [Ph. Benilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre, and J. L. Vazquez, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 22, 241-273 (1995 Zbl 0866.35037)].


35J65 Nonlinear boundary value problems for linear elliptic equations
35R70 PDEs with multivalued right-hand sides
35J20 Variational methods for second-order elliptic equations
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations


Zbl 0866.35037