DOI: 10.68381/jca25013 ISSN: 0944-6532
Some Remarks on the (Non-) Attainment of the Boundary Data for Variational Problems in the Space BV
Michael Bildhauer, Martin Fuchs
We discuss the standard relaxed version of a minimization problem for variational integrals of linear growth together with prescribed Dirichlet boundary data
u_0
u
0
and give estimates for the size of the set
\{x \in \partial \Omega: u (x) \not= u_0 (x)\}
{
x
∈
∂
Ω
:
u
(
x
)
≠
u
0
(
x
)
}
for BV-minimizers
u
u
which imply
{\cal{H}}^{n -1} \left(\left\{x \in \partial \Omega: u (x) < u_0 (x)\right\}\right) = {\cal{H}}^{n - 1} \left(\left\{x \in \partial \Omega: u (x) > u_0 (x) \right\}\right)
H
n
−
1
(
{
x
∈
∂
Ω
:
u
(
x
)
<
u
0
(
x
)
}
)
=
H
n
−
1
(
{
x
∈
∂
Ω
:
u
(
x
)
>
u
0
(
x
)
}
)
in the case of minimal surfaces
u
u
not attaining the boundary values
u_0
u
0
on a subset of
\partial \Omega
∂
Ω
with positive measure.