The Dirichlet problem for the $1$-Laplacian with a general singular term and $L^1$-data
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1-Laplacian
The Dirichlet problem for singular elliptic equations with general nonlinearities
Nel lavoro si tratta di esistenza, unicita' e regolarità per soluzioni di equazioni ellittiche singolari con termini di ordine inferiore
Elliptic problems involving the 1-Laplacian and a singular lower order term
This paper is concerned with the Dirichlet problem for an equation involving the $1$--Laplacian operator $\Delta_1 u:=\Div\left(\frac{Du}{|Du|}\right)$ and having a singular term of the type $\frac{f(x)}{u^\gamma}$. Here $f\in L^N(\Omega)$ is nonnegative, $00$ a.e., the solution satisfies those features that might be expected as well as a uniqueness result. We also give explicit 1--dimensional examples that show that, in general, uniqueness does not hold. We remark that the Anzellotti theory of $L^\infty$--divergence--measure vector fields must be extended to deal with this equation.