A strongly degenerate quasilinear equation : the elliptic case

Andreu, Fuensanta; Caselles, Vicent; Mazón, José

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 3 (2004) no. 3, pp. 555-587.

We prove existence and uniqueness of entropy solutions for the Neumann problem for the quasilinear elliptic equation $u-\mathrm{div}𝐚(u,Du)=v$, where $v\in L^1$, $𝐚(z,\xi )={\nabla }_{\xi }f(z,\xi )$, and $f$ is a convex function of $\xi$ with linear growth as $\parallel \xi \parallel \to \infty$, satisfying other additional assumptions. In particular, this class includes the case where $f(z,\xi )=\varphi \left(z\right)\psi \left(\xi \right)$, $\varphi >0$, $\psi$ being a convex function with linear growth as $\parallel \xi \parallel \to \infty$. In the second part of this work, using Crandall-Ligget’s iteration scheme, this result will permit us to prove existence and uniqueness of entropy solutions for the corresponding parabolic problem with initial data in $L^1$.

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