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Convergence of solutions to the $p$-Laplace evolution equation as $p$ goes to 1

We prove that the set of solutions to the parabolic singular $p$-Laplace equation with Dirichlet boundary conditions on a bounded Lipschitz domain $Ω$ for all space dimensions is continuous in the parameter $p\in [1,+\infty)$ and the initial data. The highly singular limit case p=1 is included. In particular, we show that the solutions $u_p$ converge strongly in $L^2(Ω)$, uniformly in time, to the solution $u_1$ of the parabolic 1-Laplace equation as $p\to 1$.