DOI: 10.1515/acv-2026-0029 ISSN: 1864-8258
The inhomogeneous total variation flow with
L
1
-data
Marta Latorre, Sergio Segura de León Abstract
This paper is devoted to the study of the Dirichlet problem for the parabolic equation driven by the 1-Laplacian operator under minimal integrability assumptions. Specifically, we consider
u
′
-
div
(
D
u
|
D
u
|
)
=
f
in
(
0
,
+
∞
)
×
Ω
,
see text
u^{\prime}-\operatorname{div}\biggl{(}\frac{Du}{|Du|}\biggr{)}=f\quad\text{in %
}(0,+\infty)\times\Omega,
where
Ω
⊂
ℝ
N
{\Omega\subset\mathbb{R}^{N}}
is a bounded open set with Lipschitz boundary and
f
∈
L
loc
1
(
0
,
+
∞
;
L
1
(
Ω
)
)
{f\in L_{\rm loc}^{1}(0,+\infty;L^{1}(\Omega))}
is the source term. The initial datum we consider belongs to
L
1
(
Ω
)
L^{1}(\Omega)
.
We establish the existence and uniqueness of entropy solutions in this low-regularity setting. Our approach relies on an approximation scheme and an entropy formulation adapted to the 1-Laplacian structure. Additional results include comparison between solutions, further regularity when data have higher integrability and an analysis of the long-time decay of solutions in the homogeneous case.