Quasi-linear elliptic problems in \(L^1\) with non homongeneous boundary conditions. (English) Zbl 1153.35026
In this paper, the following quasi-linear problem is studied
\[ \begin{aligned} -\operatorname{div} a(.,Du) + u = \varphi &\quad\text{in } \Omega,\\ \psi \in a(.,Du) \cdot\eta + \beta(u), &\quad\text{on } \partial\Omega, \end{aligned} \]
where \(\psi \in L^1(\partial\Omega)\), \(\varphi \in L^1(\Omega)\), and \(\beta\) is maximal monotone. The existence of an entropy solution in the sense of [P. Bénilan et al., Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 22, No. 2, 241–273 (1995; Zbl 0866.35037)] is established. The results rely on a variational inequality due to Browder which permits to obtain a sequence converging to a solution.
\[ \begin{aligned} -\operatorname{div} a(.,Du) + u = \varphi &\quad\text{in } \Omega,\\ \psi \in a(.,Du) \cdot\eta + \beta(u), &\quad\text{on } \partial\Omega, \end{aligned} \]
where \(\psi \in L^1(\partial\Omega)\), \(\varphi \in L^1(\Omega)\), and \(\beta\) is maximal monotone. The existence of an entropy solution in the sense of [P. Bénilan et al., Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 22, No. 2, 241–273 (1995; Zbl 0866.35037)] is established. The results rely on a variational inequality due to Browder which permits to obtain a sequence converging to a solution.
Reviewer: Marlène Frigon (Montréal)
MSC:
| 35J60 | Nonlinear elliptic equations |
| 35D05 | Existence of generalized solutions of PDE (MSC2000) |
| 35J20 | Variational methods for second-order elliptic equations |
