## $$L^1$$ 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.

### MSC:

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

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