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research
Schroedinger operators involving singular potentials and measure data
Authors
Augusto C. Ponce
Nicolas Wilmet
Publication date
1 January 2017
Publisher
'Elsevier BV'
Doi
Cite
View
on
arXiv
Abstract
We study the existence of solutions of the Dirichlet problem for the Schroedinger operator with measure data
{
−
Δ
u
+
V
u
=
μ
inÂ
Ω
,
u
=
0
onÂ
∂
Ω
.
\left\{ \begin{alignedat}{2} -\Delta u + Vu & = \mu && \quad \text{in } \Omega,\\ u & = 0 && \quad \text{on } \partial \Omega. \end{alignedat} \right.
{
−
Δ
u
+
V
u
u
​
=
μ
=
0
​
​
inÂ
Ω
,
onÂ
∂
Ω.
​
We characterize the finite measures
μ
\mu
μ
for which this problem has a solution for every nonnegative potential
V
V
V
in the Lebesgue space
L
p
(
Ω
)
L^p(\Omega)
L
p
(
Ω
)
with
1
≤
p
≤
N
/
2
1 \le p \le N/2
1
≤
p
≤
N
/2
. The full answer can be expressed in terms of the
W
2
,
p
W^{2,p}
W
2
,
p
capacity for
p
>
1
p > 1
p
>
1
, and the
W
1
,
2
W^{1,2}
W
1
,
2
(or Newtonian) capacity for
p
=
1
p = 1
p
=
1
. We then prove the existence of a solution of the problem above when
V
V
V
belongs to the real Hardy space
H
1
(
Ω
)
H^1(\Omega)
H
1
(
Ω
)
and
μ
\mu
μ
is diffuse with respect to the
W
2
,
1
W^{2,1}
W
2
,
1
capacity.Comment: Fixed a display problem in arxiv's abstract. Original tex file unchange
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Last time updated on 23/09/2018