## 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)].

### MSC:

 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