\(L^1\) existence and uniqueness results for quasi-linear elliptic equations with nonlinear boundary conditions. (English) Zbl 1123.35016

The authors study quasilinear elliptic equations with nonlinear boundary conditions of the form \[ -\text{div}\;\mathbf{a} (x, Du) + \gamma(u) \ni \phi,\;\; \text{in}\; \Omega, \]
\[ \mathbf{a} (x, Du) + \beta(u) \ni \psi,\;\; \text{on}\; \partial \Omega, \] where \(\Omega\) is an open bounded domain of \(\mathbb{R}^N\), with smooth boundary \(\partial \Omega\), \(\phi \in L^1 (\Omega)\) and \(\psi \in L^1 (\partial \Omega)\). The nonlinearities \(\gamma\), \(\beta\) are maximal monotone graphs in \(\mathbb{R}^2\), such that \(0 \in \gamma(0)\) and \(0 \in \beta(0)\). More precisely, they prove existence and uniqueness results of weak and entropy solutions in the case of general nonlinear operators \(\mathbf{a} (x, Du)\) with nonhomogeneous boundary conditions and general nonlinearities \(\beta\) and \(\gamma\). The existence results are obtained with the help of appropriate approximating problems whose solutions satisfy certain estimates and monotone properties, which allow to pass to the limit.


35J60 Nonlinear elliptic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
76D27 Other free boundary flows; Hele-Shaw flows
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