Regularity of almost minimizers with free boundary
In this paper we study the local regularity of almost minimizers of the functional J ( u ) = ∫ Ω | ∇ u ( x ) | 2 + q + 2 ( x ) χ { u > 0 } ( x ) + q - 2 ( x ) χ { u < 0 } ( x ) where q ± ∈ L ∞ ( Ω ) . Almost minimizers do not satisfy a PDE or a monotonicity formula like minimizers do (see Alt...
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| Veröffentlicht in: | Calculus of variations and partial differential equations 2015-09, Vol.54 (1), p.455-524 |
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| creator | David, G. Toro, T. |
| description | In this paper we study the local regularity of almost minimizers of the functional
J
(
u
)
=
∫
Ω
|
∇
u
(
x
)
|
2
+
q
+
2
(
x
)
χ
{
u
>
0
}
(
x
)
+
q
-
2
(
x
)
χ
{
u
<
0
}
(
x
)
where
q
±
∈
L
∞
(
Ω
)
. Almost minimizers do not satisfy a PDE or a monotonicity formula like minimizers do (see Alt and Caffarelli, in J Reine Angew Math, 325:105–144,
1981
; Alt et al., in Trans Am Math Soc 282:431–461,
1984
; Caffarelli et al., in Global energy minimizers for free boundary problems and full regularity in three dimensions. In: Non-compact Problems at the Intersection of Geometry, Analysis, and Topology, vol. 8397. Contemporary Mathematics, vol. 350. American Mathematical Society, Providence,
2004
; DeSilva and Jerison, in J Reine Angew Math 635:121,
2009
). Nevertheless we succeed in proving that they are locally Lipschitz, which is the optimal regularity for minimizers. |
| doi_str_mv | 10.1007/s00526-014-0792-z |
| format | Article |
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J
(
u
)
=
∫
Ω
|
∇
u
(
x
)
|
2
+
q
+
2
(
x
)
χ
{
u
>
0
}
(
x
)
+
q
-
2
(
x
)
χ
{
u
<
0
}
(
x
)
where
q
±
∈
L
∞
(
Ω
)
. Almost minimizers do not satisfy a PDE or a monotonicity formula like minimizers do (see Alt and Caffarelli, in J Reine Angew Math, 325:105–144,
1981
; Alt et al., in Trans Am Math Soc 282:431–461,
1984
; Caffarelli et al., in Global energy minimizers for free boundary problems and full regularity in three dimensions. In: Non-compact Problems at the Intersection of Geometry, Analysis, and Topology, vol. 8397. Contemporary Mathematics, vol. 350. American Mathematical Society, Providence,
2004
; DeSilva and Jerison, in J Reine Angew Math 635:121,
2009
). Nevertheless we succeed in proving that they are locally Lipschitz, which is the optimal regularity for minimizers.</description><identifier>ISSN: 0944-2669</identifier><identifier>EISSN: 1432-0835</identifier><identifier>DOI: 10.1007/s00526-014-0792-z</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Analysis ; Calculus of variations ; Calculus of Variations and Optimal Control; Optimization ; Control ; Formulas (mathematics) ; Free boundaries ; Mathematical analysis ; Mathematical and Computational Physics ; Mathematics ; Mathematics and Statistics ; Optimization ; Partial differential equations ; Regularity ; Systems Theory ; Texts ; Theoretical</subject><ispartof>Calculus of variations and partial differential equations, 2015-09, Vol.54 (1), p.455-524</ispartof><rights>Springer-Verlag Berlin Heidelberg 2014</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c391t-ba33d02772548d43286d120b195d482661d08648d5d8a2ce0d8944b512eb4d033</citedby><cites>FETCH-LOGICAL-c391t-ba33d02772548d43286d120b195d482661d08648d5d8a2ce0d8944b512eb4d033</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00526-014-0792-z$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00526-014-0792-z$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>David, G.</creatorcontrib><creatorcontrib>Toro, T.</creatorcontrib><title>Regularity of almost minimizers with free boundary</title><title>Calculus of variations and partial differential equations</title><addtitle>Calc. Var</addtitle><description>In this paper we study the local regularity of almost minimizers of the functional
J
(
u
)
=
∫
Ω
|
∇
u
(
x
)
|
2
+
q
+
2
(
x
)
χ
{
u
>
0
}
(
x
)
+
q
-
2
(
x
)
χ
{
u
<
0
}
(
x
)
where
q
±
∈
L
∞
(
Ω
)
. Almost minimizers do not satisfy a PDE or a monotonicity formula like minimizers do (see Alt and Caffarelli, in J Reine Angew Math, 325:105–144,
1981
; Alt et al., in Trans Am Math Soc 282:431–461,
1984
; Caffarelli et al., in Global energy minimizers for free boundary problems and full regularity in three dimensions. In: Non-compact Problems at the Intersection of Geometry, Analysis, and Topology, vol. 8397. Contemporary Mathematics, vol. 350. American Mathematical Society, Providence,
2004
; DeSilva and Jerison, in J Reine Angew Math 635:121,
2009
). Nevertheless we succeed in proving that they are locally Lipschitz, which is the optimal regularity for minimizers.</description><subject>Analysis</subject><subject>Calculus of variations</subject><subject>Calculus of Variations and Optimal Control; Optimization</subject><subject>Control</subject><subject>Formulas (mathematics)</subject><subject>Free boundaries</subject><subject>Mathematical analysis</subject><subject>Mathematical and Computational Physics</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Optimization</subject><subject>Partial differential equations</subject><subject>Regularity</subject><subject>Systems Theory</subject><subject>Texts</subject><subject>Theoretical</subject><issn>0944-2669</issn><issn>1432-0835</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2015</creationdate><recordtype>article</recordtype><recordid>eNp9kMtOwzAQRS0EEqXwAeyyZGOY8SNxlqgCilQJCcHacmKnuMqj2IlQ-_W4CmtWs5h7R2cOIbcI9whQPEQAyXIKKCgUJaPHM7JAwRkFxeU5WUApBGV5Xl6Sqxh3ACgVEwvC3t12ak3w4yEbmsy03RDHrPO97_zRhZj9-PEra4JzWTVMvTXhcE0uGtNGd_M3l-Tz-eljtaabt5fX1eOG1rzEkVaGcwusKJgUyiYUlVtkUGEprVAJBS2oPK2kVYbVDqxKjJVE5iphgfMluZvv7sPwPbk46s7H2rWt6d0wRY0lCIYFZ5iiOEfrMMQYXKP3wXeJVSPokx89-9HJjz750cfUYXMnpmy_dUHvhin06aN_Sr8SE2cu</recordid><startdate>20150901</startdate><enddate>20150901</enddate><creator>David, G.</creator><creator>Toro, T.</creator><general>Springer Berlin Heidelberg</general><scope>AAYXX</scope><scope>CITATION</scope><scope>7TB</scope><scope>7U5</scope><scope>8FD</scope><scope>FR3</scope><scope>H8D</scope><scope>KR7</scope><scope>L7M</scope></search><sort><creationdate>20150901</creationdate><title>Regularity of almost minimizers with free boundary</title><author>David, G. ; Toro, T.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c391t-ba33d02772548d43286d120b195d482661d08648d5d8a2ce0d8944b512eb4d033</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2015</creationdate><topic>Analysis</topic><topic>Calculus of variations</topic><topic>Calculus of Variations and Optimal Control; Optimization</topic><topic>Control</topic><topic>Formulas (mathematics)</topic><topic>Free boundaries</topic><topic>Mathematical analysis</topic><topic>Mathematical and Computational Physics</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Optimization</topic><topic>Partial differential equations</topic><topic>Regularity</topic><topic>Systems Theory</topic><topic>Texts</topic><topic>Theoretical</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>David, G.</creatorcontrib><creatorcontrib>Toro, T.</creatorcontrib><collection>CrossRef</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>Solid State and Superconductivity Abstracts</collection><collection>Technology Research Database</collection><collection>Engineering Research Database</collection><collection>Aerospace Database</collection><collection>Civil Engineering Abstracts</collection><collection>Advanced Technologies Database with Aerospace</collection><jtitle>Calculus of variations and partial differential equations</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>David, G.</au><au>Toro, T.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Regularity of almost minimizers with free boundary</atitle><jtitle>Calculus of variations and partial differential equations</jtitle><stitle>Calc. Var</stitle><date>2015-09-01</date><risdate>2015</risdate><volume>54</volume><issue>1</issue><spage>455</spage><epage>524</epage><pages>455-524</pages><issn>0944-2669</issn><eissn>1432-0835</eissn><abstract>In this paper we study the local regularity of almost minimizers of the functional
J
(
u
)
=
∫
Ω
|
∇
u
(
x
)
|
2
+
q
+
2
(
x
)
χ
{
u
>
0
}
(
x
)
+
q
-
2
(
x
)
χ
{
u
<
0
}
(
x
)
where
q
±
∈
L
∞
(
Ω
)
. Almost minimizers do not satisfy a PDE or a monotonicity formula like minimizers do (see Alt and Caffarelli, in J Reine Angew Math, 325:105–144,
1981
; Alt et al., in Trans Am Math Soc 282:431–461,
1984
; Caffarelli et al., in Global energy minimizers for free boundary problems and full regularity in three dimensions. In: Non-compact Problems at the Intersection of Geometry, Analysis, and Topology, vol. 8397. Contemporary Mathematics, vol. 350. American Mathematical Society, Providence,
2004
; DeSilva and Jerison, in J Reine Angew Math 635:121,
2009
). Nevertheless we succeed in proving that they are locally Lipschitz, which is the optimal regularity for minimizers.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s00526-014-0792-z</doi><tpages>70</tpages></addata></record> |
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| source | SpringerLink Journals - AutoHoldings |
| subjects | Analysis Calculus of variations Calculus of Variations and Optimal Control Optimization Control Formulas (mathematics) Free boundaries Mathematical analysis Mathematical and Computational Physics Mathematics Mathematics and Statistics Optimization Partial differential equations Regularity Systems Theory Texts Theoretical |
| title | Regularity of almost minimizers with free boundary |
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