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research
Elliptic problems with growth in nonreflexive Orlicz spaces and with measure or
L
1
L^1
L
1
data
Authors
Iwona Chlebicka
Flavia Giannetti
Anna Zatorska-Goldstein
Publication date
2 August 2018
Publisher
View
on
arXiv
Abstract
We investigate solutions to nonlinear elliptic Dirichlet problems of the type
{
−
d
i
v
A
(
x
,
u
,
∇
u
)
=
μ
i
n
Ω
,
u
=
0
o
n
∂
Ω
,
\left\{\begin{array}{cl} - {\rm div} A(x,u,\nabla u)= \mu &\qquad \mathrm{ in}\qquad \Omega, u=0 &\qquad \mathrm{ on}\qquad \partial\Omega, \end{array}\right.
{
−
div
A
(
x
,
u
,
∇
u
)
=
μ
​
in
Ω
,
u
=
0
​
on
∂
Ω
,
​
where
Ω
\Omega
Ω
is a bounded Lipschitz domain in
R
n
\mathbb{R}^n
R
n
and
A
(
x
,
z
,
ξ
)
A(x,z,\xi)
A
(
x
,
z
,
ξ
)
is a Carath\'eodory's function. The growth of~the~monotone vector field
A
A
A
with respect to the
(
z
,
ξ
)
(z,\xi)
(
z
,
ξ
)
variables is expressed through some
N
N
N
-functions
B
B
B
and
P
P
P
. We do not require any particular type of growth condition of such functions, so we deal with problems in nonreflexive spaces. When the problem involves measure data and weakly monotone operator, we prove existence. For
L
1
L^1
L
1
-data problems with strongly monotone operator we infer also uniqueness and regularity of~solutions and their gradients in the scale of Orlicz-Marcinkiewicz spaces
Similar works
Full text
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Archivio della ricerca - Università degli studi di Napoli Federico II
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Last time updated on 05/11/2019