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We present first results on the Dirichlet-to-Neumann operator associated withthe $1$-Laplace operator in $L^1$. In particular, we show that this operatorcan be realized as a sub-differential operator in $L^1\times L^{\infty}$ of ahomogeneous convex, continuous functional with effective domain $L^1$. Eventhough the Dirichlet problem associated with the $1$-Laplace operator loses theproperty that weak solutions for boundary data in $L^1$ are unique, we prove atype of stability/compactness result with respect to the boundary data in $L^1$of this problem. We apply our results for the stationary Dirichlet problem toevolution problems governed by the Dirichlet-to-Neumann operator, which canequivalently be formulated as singular coupled elliptic-parabolic initialboundary-value problems. For initial data in $L^q$, $1\le q\le \infty$, weobtain well-posedness, that every mild solution is, indeed, a strong solution,and establish long-time stability of the semigroup generated by the negativeDirichlet-to-Neumann operator associated with the $1$-Laplace operator.

The Dirichlet-to-Neumann operator associated with the $1$-Laplace operator and evolution problems