DOI: 10.68381/jca18059 ISSN: 0944-6532
About the Regularity of Average Distance Minimizers in ℝ
2
Antoine Lemenant
We focus on the following irrigation problem introduced by G. Buttazzo, E. Oudet and E. Stepanov [Optimal transportation problems with free Dirichlet regions, in: Variational Methods for Discontinuous Structures, Progr. Nonlinear Differential Equations Appl. 51, Birkhäuser, Basel (2002) 41–65]:
\min \mathcal{F}(\Sigma):=\int_{\Omega}dist(x,\Sigma)\; \mathrm{d}\mu(x),
min
F
(
Σ
)
:
=
∫
Ω
d
i
s
t
(
x
,
Σ
)
d
μ
(
x
)
,
where
\Omega
Ω
is an open subset of
\mathbb R^2
R
2
,
\mu
μ
is a probability measure and where the minimum is taken over all the sets
\Sigma \subset \Omega
Σ
⊂
Ω
such that
\Sigma
Σ
is compact, connected, and
\mathcal {H}^{1}(\Sigma)\leq \alpha_0
H
1
(
Σ
)
≤
α
0
for a given positive constant
\alpha_0
α
0
. In this paper we seek for some conditions to find in
\Sigma
Σ
some pieces of
C^1
C
1
(or more) regular curves. We prove that it is the case in the ball
B
B
when
\Sigma \cap B
Σ
∩
B
contains no corner points. More generally we prove that the Left and Right tangents half lines of
\Sigma
Σ
(that exist everywhere out of endpoints and triple points) are semicontinuous. We also discuss how the regularity is linked with the pull back measure
\psi:= k \sharp \mu
ψ
:
=
k
♯
μ
where
k
k
is the projection on
\Sigma
Σ
. In particular
\Sigma \cap B
Σ
∩
B
is
C^{1,\alpha}
C
1
,
α
when
\psi
ψ
is regular with respect to
\mathcal {H}^1
H
1
with density in a certain
L^p
L
p
. We also prove that
\Sigma
Σ
is locally a Lipschitz graph away from triple points and endpoints, and that the mean curvature of
\Sigma
Σ
is a measure that is explicited in terms of measure
\psi
ψ