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On the Regularity of the Minimizer of the Electrostatic Born–Infeld Energy
We consider the electrostatic Born–Infeld energy ∫ R N 1 - 1 - | ∇ u | 2 d x - ∫ R N ρ u d x , where ρ ∈ L m ( R N ) is an assigned charge density, m ∈ [ 1 , 2 ∗ ] , 2 ∗ : = 2 N N + 2 , N ≥ 3 . We prove that if ρ ∈ L q ( R N ) for q > 2 N , the unique minimizer u ρ is of class W loc 2 , 2 ( R N )...
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| Published in: | Archive for rational mechanics and analysis 2019-05, Vol.232 (2), p.697-725 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
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| cited_by | cdi_FETCH-LOGICAL-c316t-86fa753e2a05102f8093553985144b34ef3723a98a23eae788aab56fa9067063 |
|---|---|
| cites | cdi_FETCH-LOGICAL-c316t-86fa753e2a05102f8093553985144b34ef3723a98a23eae788aab56fa9067063 |
| container_end_page | 725 |
| container_issue | 2 |
| container_start_page | 697 |
| container_title | Archive for rational mechanics and analysis |
| container_volume | 232 |
| creator | Bonheure, Denis Iacopetti, Alessandro |
| description | We consider the electrostatic Born–Infeld energy
∫
R
N
1
-
1
-
|
∇
u
|
2
d
x
-
∫
R
N
ρ
u
d
x
,
where
ρ
∈
L
m
(
R
N
)
is an assigned charge density,
m
∈
[
1
,
2
∗
]
,
2
∗
:
=
2
N
N
+
2
,
N
≥
3
. We prove that if
ρ
∈
L
q
(
R
N
)
for
q
>
2
N
, the unique minimizer
u
ρ
is of class
W
loc
2
,
2
(
R
N
)
. Moreover, if the norm of
ρ
is sufficiently small, the minimizer is a weak solution of the associated PDE
-
div
∇
u
1
-
|
∇
u
|
2
=
ρ
in
R
N
,
(
BI
)
with the boundary condition
lim
|
x
|
→
∞
u
(
x
)
=
0
, and it is of class
C
loc
1
,
α
(
R
N
)
for some
α
∈
(
0
,
1
)
. |
| doi_str_mv | 10.1007/s00205-018-1331-4 |
| format | article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_2194643412</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2194643412</sourcerecordid><originalsourceid>FETCH-LOGICAL-c316t-86fa753e2a05102f8093553985144b34ef3723a98a23eae788aab56fa9067063</originalsourceid><addsrcrecordid>eNp1kEFOwzAQRS0EEqVwAHaRWBvGHjtxllCVUqmoEurecoNdUqVJsd1FWXEHbshJcAmIFavRjP7_M_MIuWRwzQCKmwDAQVJgijJERsURGTCBnEJe4DEZAADSUvLilJyFsD60HPMBmc3bLL7Y7Mmudo3xddxnnfuePNZtvanfrP8djBtbRd-FaGJdZXedbz_fP6ats81zNm6tX-3PyYkzTbAXP3VIFvfjxeiBzuaT6eh2RitkeaQqd6aQaLkByYA7BSVKiaWSTIglCuuw4GhKZThaYwuljFnKZCrTM5DjkFz1sVvfve5siHrd7XybNmrOSpELFIwnFetVVbo5eOv01tcb4_eagT4w0z0znZjpAzMtkof3npC07cr6v-T_TV-1K22d</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2194643412</pqid></control><display><type>article</type><title>On the Regularity of the Minimizer of the Electrostatic Born–Infeld Energy</title><source>Springer Nature</source><creator>Bonheure, Denis ; Iacopetti, Alessandro</creator><creatorcontrib>Bonheure, Denis ; Iacopetti, Alessandro</creatorcontrib><description>We consider the electrostatic Born–Infeld energy
∫
R
N
1
-
1
-
|
∇
u
|
2
d
x
-
∫
R
N
ρ
u
d
x
,
where
ρ
∈
L
m
(
R
N
)
is an assigned charge density,
m
∈
[
1
,
2
∗
]
,
2
∗
:
=
2
N
N
+
2
,
N
≥
3
. We prove that if
ρ
∈
L
q
(
R
N
)
for
q
>
2
N
, the unique minimizer
u
ρ
is of class
W
loc
2
,
2
(
R
N
)
. Moreover, if the norm of
ρ
is sufficiently small, the minimizer is a weak solution of the associated PDE
-
div
∇
u
1
-
|
∇
u
|
2
=
ρ
in
R
N
,
(
BI
)
with the boundary condition
lim
|
x
|
→
∞
u
(
x
)
=
0
, and it is of class
C
loc
1
,
α
(
R
N
)
for some
α
∈
(
0
,
1
)
.</description><identifier>ISSN: 0003-9527</identifier><identifier>EISSN: 1432-0673</identifier><identifier>DOI: 10.1007/s00205-018-1331-4</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Boundary conditions ; Charge density ; Classical Mechanics ; Complex Systems ; Fluid- and Aerodynamics ; Mathematical and Computational Physics ; Physics ; Physics and Astronomy ; Theoretical</subject><ispartof>Archive for rational mechanics and analysis, 2019-05, Vol.232 (2), p.697-725</ispartof><rights>Springer-Verlag GmbH Germany, part of Springer Nature 2018</rights><rights>Copyright Springer Nature B.V. 2019</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c316t-86fa753e2a05102f8093553985144b34ef3723a98a23eae788aab56fa9067063</citedby><cites>FETCH-LOGICAL-c316t-86fa753e2a05102f8093553985144b34ef3723a98a23eae788aab56fa9067063</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,776,780,27901,27902</link.rule.ids></links><search><creatorcontrib>Bonheure, Denis</creatorcontrib><creatorcontrib>Iacopetti, Alessandro</creatorcontrib><title>On the Regularity of the Minimizer of the Electrostatic Born–Infeld Energy</title><title>Archive for rational mechanics and analysis</title><addtitle>Arch Rational Mech Anal</addtitle><description>We consider the electrostatic Born–Infeld energy
∫
R
N
1
-
1
-
|
∇
u
|
2
d
x
-
∫
R
N
ρ
u
d
x
,
where
ρ
∈
L
m
(
R
N
)
is an assigned charge density,
m
∈
[
1
,
2
∗
]
,
2
∗
:
=
2
N
N
+
2
,
N
≥
3
. We prove that if
ρ
∈
L
q
(
R
N
)
for
q
>
2
N
, the unique minimizer
u
ρ
is of class
W
loc
2
,
2
(
R
N
)
. Moreover, if the norm of
ρ
is sufficiently small, the minimizer is a weak solution of the associated PDE
-
div
∇
u
1
-
|
∇
u
|
2
=
ρ
in
R
N
,
(
BI
)
with the boundary condition
lim
|
x
|
→
∞
u
(
x
)
=
0
, and it is of class
C
loc
1
,
α
(
R
N
)
for some
α
∈
(
0
,
1
)
.</description><subject>Boundary conditions</subject><subject>Charge density</subject><subject>Classical Mechanics</subject><subject>Complex Systems</subject><subject>Fluid- and Aerodynamics</subject><subject>Mathematical and Computational Physics</subject><subject>Physics</subject><subject>Physics and Astronomy</subject><subject>Theoretical</subject><issn>0003-9527</issn><issn>1432-0673</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2019</creationdate><recordtype>article</recordtype><recordid>eNp1kEFOwzAQRS0EEqVwAHaRWBvGHjtxllCVUqmoEurecoNdUqVJsd1FWXEHbshJcAmIFavRjP7_M_MIuWRwzQCKmwDAQVJgijJERsURGTCBnEJe4DEZAADSUvLilJyFsD60HPMBmc3bLL7Y7Mmudo3xddxnnfuePNZtvanfrP8djBtbRd-FaGJdZXedbz_fP6ats81zNm6tX-3PyYkzTbAXP3VIFvfjxeiBzuaT6eh2RitkeaQqd6aQaLkByYA7BSVKiaWSTIglCuuw4GhKZThaYwuljFnKZCrTM5DjkFz1sVvfve5siHrd7XybNmrOSpELFIwnFetVVbo5eOv01tcb4_eagT4w0z0znZjpAzMtkof3npC07cr6v-T_TV-1K22d</recordid><startdate>20190502</startdate><enddate>20190502</enddate><creator>Bonheure, Denis</creator><creator>Iacopetti, Alessandro</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>7TB</scope><scope>8FD</scope><scope>FR3</scope><scope>JQ2</scope><scope>KR7</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope></search><sort><creationdate>20190502</creationdate><title>On the Regularity of the Minimizer of the Electrostatic Born–Infeld Energy</title><author>Bonheure, Denis ; Iacopetti, Alessandro</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c316t-86fa753e2a05102f8093553985144b34ef3723a98a23eae788aab56fa9067063</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2019</creationdate><topic>Boundary conditions</topic><topic>Charge density</topic><topic>Classical Mechanics</topic><topic>Complex Systems</topic><topic>Fluid- and Aerodynamics</topic><topic>Mathematical and Computational Physics</topic><topic>Physics</topic><topic>Physics and Astronomy</topic><topic>Theoretical</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Bonheure, Denis</creatorcontrib><creatorcontrib>Iacopetti, Alessandro</creatorcontrib><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>Technology Research Database</collection><collection>Engineering Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Civil Engineering Abstracts</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Archive for rational mechanics and analysis</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Bonheure, Denis</au><au>Iacopetti, Alessandro</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On the Regularity of the Minimizer of the Electrostatic Born–Infeld Energy</atitle><jtitle>Archive for rational mechanics and analysis</jtitle><stitle>Arch Rational Mech Anal</stitle><date>2019-05-02</date><risdate>2019</risdate><volume>232</volume><issue>2</issue><spage>697</spage><epage>725</epage><pages>697-725</pages><issn>0003-9527</issn><eissn>1432-0673</eissn><abstract>We consider the electrostatic Born–Infeld energy
∫
R
N
1
-
1
-
|
∇
u
|
2
d
x
-
∫
R
N
ρ
u
d
x
,
where
ρ
∈
L
m
(
R
N
)
is an assigned charge density,
m
∈
[
1
,
2
∗
]
,
2
∗
:
=
2
N
N
+
2
,
N
≥
3
. We prove that if
ρ
∈
L
q
(
R
N
)
for
q
>
2
N
, the unique minimizer
u
ρ
is of class
W
loc
2
,
2
(
R
N
)
. Moreover, if the norm of
ρ
is sufficiently small, the minimizer is a weak solution of the associated PDE
-
div
∇
u
1
-
|
∇
u
|
2
=
ρ
in
R
N
,
(
BI
)
with the boundary condition
lim
|
x
|
→
∞
u
(
x
)
=
0
, and it is of class
C
loc
1
,
α
(
R
N
)
for some
α
∈
(
0
,
1
)
.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s00205-018-1331-4</doi><tpages>29</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0003-9527 |
| ispartof | Archive for rational mechanics and analysis, 2019-05, Vol.232 (2), p.697-725 |
| issn | 0003-9527 1432-0673 |
| language | eng |
| recordid | cdi_proquest_journals_2194643412 |
| source | Springer Nature |
| subjects | Boundary conditions Charge density Classical Mechanics Complex Systems Fluid- and Aerodynamics Mathematical and Computational Physics Physics Physics and Astronomy Theoretical |
| title | On the Regularity of the Minimizer of the Electrostatic Born–Infeld Energy |
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