A comparison result related to Gauss measure
In this paper we prove a comparison result for weak solutions to linear elliptic problems of the type −(a ij(x)u x i ) x j =f(x)ϕ(x) in Ω, u=0 on ∂Ω, where Ω is an open set of R n ( n⩾2), ϕ( x)=(2 π) − n/2 exp(−| x| 2/2), a ij ( x) are measurable functions such that a ij ( x) ξ i ξ j ⩾ ϕ( x)| ξ| 2 a...
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| Veröffentlicht in: | Comptes rendus. Mathématique 2002-01, Vol.334 (6), p.451-456 |
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| creator | Betta, M.Francesca Brock, Friedman Mercaldo, Anna Posteraro, M.Rosaria |
| description | In this paper we prove a comparison result for weak solutions to linear elliptic problems of the type
−(a
ij(x)u
x
i
)
x
j
=f(x)ϕ(x)
in
Ω,
u=0
on
∂Ω,
where
Ω is an open set of
R
n
(
n⩾2),
ϕ(
x)=(2
π)
−
n/2
exp(−|
x|
2/2),
a
ij
(
x) are measurable functions such that
a
ij
(
x)
ξ
i
ξ
j
⩾
ϕ(
x)|
ξ|
2 a.e.
x∈Ω,
ξ∈
R
n
and
f(
x) is a measurable function taken in order to guarantee the existence of a solution
u∈
H
1
0(ϕ,Ω)
of (1.1). We use the notion of rearrangement related to Gauss measure to compare
u(
x) with the solution of a problem of the same type, whose data are defined in a half-space and depend only on one variable.
To cite this article: M.F. Betta et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 451–456.
Dans cette note on démontre un résultat de comparaison pour les solutions faibles de problèmes elliptiques linéaires du type
−(a
ij(x)u
x
i
)
x
j
=f(x)ϕ(x)
dans
Ω,
u=0
sur
∂Ω,
où
Ω est un ouvert de
R
n
(
n⩾2),
ϕ(
x)=(2
π)
−
n/2
exp(−|
x|
2/2),
a
ij
(
x) sont des fonctions mesurables telles que
a
ij
(
x)
ξ
i
ξ
j
⩾
ϕ(
x)|
ξ|
2 p.p.
x∈Ω,
ξ∈
R
n
et
f(
x) est une fonction mesurable telle qu'il existe une solution
u de (0.1), dans
H
1
0(ϕ,Ω)
. On utilise la notion de rearrangement relatif à la mesure de Gauss pour comparer
u(
x) avec la solution d'un problème du même type, dont les données sont définies dans un demi plan et dépendent d'une variable seulement.
Pour citer cet article : M.F. Betta et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 451–456. |
| doi_str_mv | 10.1016/S1631-073X(02)02295-1 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_miscellaneous_32282927</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><els_id>S1631073X02022951</els_id><sourcerecordid>32282927</sourcerecordid><originalsourceid>FETCH-LOGICAL-c347t-f9ac57632cebdd8f14f9e90921b760d2a71793d88250ee5362b1691ff6b3c97b3</originalsourceid><addsrcrecordid>eNqFkElLBDEQhYMoOI7-BKEvioKtWaaTzkmGwQ0GPKjgLaTTFYj0Mqa6Bf-9mUU8eqo6fK9evUfIKaPXjDJ588KkYDlV4v2C8kvKuS5ytkcmTKkyF4XU-2n_RQ7JEeIHTTqt9IRczTPXtysbA_ZdFgHHZkijsQPU2dBnD3ZEzFqwOEY4JgfeNggnuzklb_d3r4vHfPn88LSYL3MnZmrIvbauUFJwB1Vdl57NvAZNNWeVkrTmVjGlRV2WvKAAhZC8St8w72UlnFaVmJLz7d1V7D9HwMG0AR00je2gH9EIzkuuuUpgsQVd7BEjeLOKobXx2zBq1t2YTTdmHdxQbjbdGJZ0ZzsDi842PtrOBfwTi0KIUtDE3W45SGm_AkSDLkDnoA4R3GDqPvzj9ANy_3dt</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>32282927</pqid></control><display><type>article</type><title>A comparison result related to Gauss measure</title><source>Elsevier ScienceDirect Journals</source><source>Elektronische Zeitschriftenbibliothek - Frei zugängliche E-Journals</source><creator>Betta, M.Francesca ; Brock, Friedman ; Mercaldo, Anna ; Posteraro, M.Rosaria</creator><creatorcontrib>Betta, M.Francesca ; Brock, Friedman ; Mercaldo, Anna ; Posteraro, M.Rosaria</creatorcontrib><description>In this paper we prove a comparison result for weak solutions to linear elliptic problems of the type
−(a
ij(x)u
x
i
)
x
j
=f(x)ϕ(x)
in
Ω,
u=0
on
∂Ω,
where
Ω is an open set of
R
n
(
n⩾2),
ϕ(
x)=(2
π)
−
n/2
exp(−|
x|
2/2),
a
ij
(
x) are measurable functions such that
a
ij
(
x)
ξ
i
ξ
j
⩾
ϕ(
x)|
ξ|
2 a.e.
x∈Ω,
ξ∈
R
n
and
f(
x) is a measurable function taken in order to guarantee the existence of a solution
u∈
H
1
0(ϕ,Ω)
of (1.1). We use the notion of rearrangement related to Gauss measure to compare
u(
x) with the solution of a problem of the same type, whose data are defined in a half-space and depend only on one variable.
To cite this article: M.F. Betta et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 451–456.
Dans cette note on démontre un résultat de comparaison pour les solutions faibles de problèmes elliptiques linéaires du type
−(a
ij(x)u
x
i
)
x
j
=f(x)ϕ(x)
dans
Ω,
u=0
sur
∂Ω,
où
Ω est un ouvert de
R
n
(
n⩾2),
ϕ(
x)=(2
π)
−
n/2
exp(−|
x|
2/2),
a
ij
(
x) sont des fonctions mesurables telles que
a
ij
(
x)
ξ
i
ξ
j
⩾
ϕ(
x)|
ξ|
2 p.p.
x∈Ω,
ξ∈
R
n
et
f(
x) est une fonction mesurable telle qu'il existe une solution
u de (0.1), dans
H
1
0(ϕ,Ω)
. On utilise la notion de rearrangement relatif à la mesure de Gauss pour comparer
u(
x) avec la solution d'un problème du même type, dont les données sont définies dans un demi plan et dépendent d'une variable seulement.
Pour citer cet article : M.F. Betta et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 451–456.</description><identifier>ISSN: 1631-073X</identifier><identifier>ISSN: 1778-3569</identifier><identifier>EISSN: 1778-3569</identifier><identifier>DOI: 10.1016/S1631-073X(02)02295-1</identifier><language>eng</language><publisher>Paris: Elsevier SAS</publisher><subject>Exact sciences and technology ; Mathematical analysis ; Mathematics ; Partial differential equations ; Sciences and techniques of general use</subject><ispartof>Comptes rendus. Mathématique, 2002-01, Vol.334 (6), p.451-456</ispartof><rights>2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS</rights><rights>2002 INIST-CNRS</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c347t-f9ac57632cebdd8f14f9e90921b760d2a71793d88250ee5362b1691ff6b3c97b3</citedby><cites>FETCH-LOGICAL-c347t-f9ac57632cebdd8f14f9e90921b760d2a71793d88250ee5362b1691ff6b3c97b3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://www.sciencedirect.com/science/article/pii/S1631073X02022951$$EHTML$$P50$$Gelsevier$$H</linktohtml><link.rule.ids>314,776,780,3536,27903,27904,65309</link.rule.ids><backlink>$$Uhttp://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=13533830$$DView record in Pascal Francis$$Hfree_for_read</backlink></links><search><creatorcontrib>Betta, M.Francesca</creatorcontrib><creatorcontrib>Brock, Friedman</creatorcontrib><creatorcontrib>Mercaldo, Anna</creatorcontrib><creatorcontrib>Posteraro, M.Rosaria</creatorcontrib><title>A comparison result related to Gauss measure</title><title>Comptes rendus. Mathématique</title><description>In this paper we prove a comparison result for weak solutions to linear elliptic problems of the type
−(a
ij(x)u
x
i
)
x
j
=f(x)ϕ(x)
in
Ω,
u=0
on
∂Ω,
where
Ω is an open set of
R
n
(
n⩾2),
ϕ(
x)=(2
π)
−
n/2
exp(−|
x|
2/2),
a
ij
(
x) are measurable functions such that
a
ij
(
x)
ξ
i
ξ
j
⩾
ϕ(
x)|
ξ|
2 a.e.
x∈Ω,
ξ∈
R
n
and
f(
x) is a measurable function taken in order to guarantee the existence of a solution
u∈
H
1
0(ϕ,Ω)
of (1.1). We use the notion of rearrangement related to Gauss measure to compare
u(
x) with the solution of a problem of the same type, whose data are defined in a half-space and depend only on one variable.
To cite this article: M.F. Betta et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 451–456.
Dans cette note on démontre un résultat de comparaison pour les solutions faibles de problèmes elliptiques linéaires du type
−(a
ij(x)u
x
i
)
x
j
=f(x)ϕ(x)
dans
Ω,
u=0
sur
∂Ω,
où
Ω est un ouvert de
R
n
(
n⩾2),
ϕ(
x)=(2
π)
−
n/2
exp(−|
x|
2/2),
a
ij
(
x) sont des fonctions mesurables telles que
a
ij
(
x)
ξ
i
ξ
j
⩾
ϕ(
x)|
ξ|
2 p.p.
x∈Ω,
ξ∈
R
n
et
f(
x) est une fonction mesurable telle qu'il existe une solution
u de (0.1), dans
H
1
0(ϕ,Ω)
. On utilise la notion de rearrangement relatif à la mesure de Gauss pour comparer
u(
x) avec la solution d'un problème du même type, dont les données sont définies dans un demi plan et dépendent d'une variable seulement.
Pour citer cet article : M.F. Betta et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 451–456.</description><subject>Exact sciences and technology</subject><subject>Mathematical analysis</subject><subject>Mathematics</subject><subject>Partial differential equations</subject><subject>Sciences and techniques of general use</subject><issn>1631-073X</issn><issn>1778-3569</issn><issn>1778-3569</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2002</creationdate><recordtype>article</recordtype><recordid>eNqFkElLBDEQhYMoOI7-BKEvioKtWaaTzkmGwQ0GPKjgLaTTFYj0Mqa6Bf-9mUU8eqo6fK9evUfIKaPXjDJ588KkYDlV4v2C8kvKuS5ytkcmTKkyF4XU-2n_RQ7JEeIHTTqt9IRczTPXtysbA_ZdFgHHZkijsQPU2dBnD3ZEzFqwOEY4JgfeNggnuzklb_d3r4vHfPn88LSYL3MnZmrIvbauUFJwB1Vdl57NvAZNNWeVkrTmVjGlRV2WvKAAhZC8St8w72UlnFaVmJLz7d1V7D9HwMG0AR00je2gH9EIzkuuuUpgsQVd7BEjeLOKobXx2zBq1t2YTTdmHdxQbjbdGJZ0ZzsDi842PtrOBfwTi0KIUtDE3W45SGm_AkSDLkDnoA4R3GDqPvzj9ANy_3dt</recordid><startdate>20020101</startdate><enddate>20020101</enddate><creator>Betta, M.Francesca</creator><creator>Brock, Friedman</creator><creator>Mercaldo, Anna</creator><creator>Posteraro, M.Rosaria</creator><general>Elsevier SAS</general><general>Elsevier</general><scope>IQODW</scope><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>20020101</creationdate><title>A comparison result related to Gauss measure</title><author>Betta, M.Francesca ; Brock, Friedman ; Mercaldo, Anna ; Posteraro, M.Rosaria</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c347t-f9ac57632cebdd8f14f9e90921b760d2a71793d88250ee5362b1691ff6b3c97b3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2002</creationdate><topic>Exact sciences and technology</topic><topic>Mathematical analysis</topic><topic>Mathematics</topic><topic>Partial differential equations</topic><topic>Sciences and techniques of general use</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Betta, M.Francesca</creatorcontrib><creatorcontrib>Brock, Friedman</creatorcontrib><creatorcontrib>Mercaldo, Anna</creatorcontrib><creatorcontrib>Posteraro, M.Rosaria</creatorcontrib><collection>Pascal-Francis</collection><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>Comptes rendus. Mathématique</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Betta, M.Francesca</au><au>Brock, Friedman</au><au>Mercaldo, Anna</au><au>Posteraro, M.Rosaria</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>A comparison result related to Gauss measure</atitle><jtitle>Comptes rendus. Mathématique</jtitle><date>2002-01-01</date><risdate>2002</risdate><volume>334</volume><issue>6</issue><spage>451</spage><epage>456</epage><pages>451-456</pages><issn>1631-073X</issn><issn>1778-3569</issn><eissn>1778-3569</eissn><abstract>In this paper we prove a comparison result for weak solutions to linear elliptic problems of the type
−(a
ij(x)u
x
i
)
x
j
=f(x)ϕ(x)
in
Ω,
u=0
on
∂Ω,
where
Ω is an open set of
R
n
(
n⩾2),
ϕ(
x)=(2
π)
−
n/2
exp(−|
x|
2/2),
a
ij
(
x) are measurable functions such that
a
ij
(
x)
ξ
i
ξ
j
⩾
ϕ(
x)|
ξ|
2 a.e.
x∈Ω,
ξ∈
R
n
and
f(
x) is a measurable function taken in order to guarantee the existence of a solution
u∈
H
1
0(ϕ,Ω)
of (1.1). We use the notion of rearrangement related to Gauss measure to compare
u(
x) with the solution of a problem of the same type, whose data are defined in a half-space and depend only on one variable.
To cite this article: M.F. Betta et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 451–456.
Dans cette note on démontre un résultat de comparaison pour les solutions faibles de problèmes elliptiques linéaires du type
−(a
ij(x)u
x
i
)
x
j
=f(x)ϕ(x)
dans
Ω,
u=0
sur
∂Ω,
où
Ω est un ouvert de
R
n
(
n⩾2),
ϕ(
x)=(2
π)
−
n/2
exp(−|
x|
2/2),
a
ij
(
x) sont des fonctions mesurables telles que
a
ij
(
x)
ξ
i
ξ
j
⩾
ϕ(
x)|
ξ|
2 p.p.
x∈Ω,
ξ∈
R
n
et
f(
x) est une fonction mesurable telle qu'il existe une solution
u de (0.1), dans
H
1
0(ϕ,Ω)
. On utilise la notion de rearrangement relatif à la mesure de Gauss pour comparer
u(
x) avec la solution d'un problème du même type, dont les données sont définies dans un demi plan et dépendent d'une variable seulement.
Pour citer cet article : M.F. Betta et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 451–456.</abstract><cop>Paris</cop><pub>Elsevier SAS</pub><doi>10.1016/S1631-073X(02)02295-1</doi><tpages>6</tpages><oa>free_for_read</oa></addata></record> |
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| issn | 1631-073X 1778-3569 1778-3569 |
| language | eng |
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| source | Elsevier ScienceDirect Journals; Elektronische Zeitschriftenbibliothek - Frei zugängliche E-Journals |
| subjects | Exact sciences and technology Mathematical analysis Mathematics Partial differential equations Sciences and techniques of general use |
| title | A comparison result related to Gauss measure |
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