This is the abstract of the paper
  "On the generation of $C_0$-semigroups in $L^1(I)$"
by A. Attalienti and M. Campiti.

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Abstract: In this paper we characterize
completely the existence of a $C_0$-semigroup
in $L^1(I)$ ($I$ real interval) generated
by a second-order differential operator when suitable
boundary conditions at the endpoints are imposed.
In spaces of continuous functions similar characterizations
have been obtained by Timmermans on the maximal domain,
by Cl\'ement and Timmermans in the
case of Ventcel's boundary conditions and
by Campiti, Metafune and Pallara in the case
of Neumann's boundary conditions.