The sharp upper bound for the area of the nodal sets of Dirichlet Laplace eigenfunctions
Let Ω be a bounded domain in R n with C 1 boundary and let u λ be a Dirichlet Laplace eigenfunction in Ω with eigenvalue λ . We show that the ( n - 1 ) -dimensional Hausdorff measure of the zero set of u λ does not exceed C ( Ω ) λ . This result is new even for the case of domains with C ∞ -smooth b...
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| Veröffentlicht in: | Geometric and functional analysis 2021-10, Vol.31 (5), p.1219-1244 |
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| container_title | Geometric and functional analysis |
| container_volume | 31 |
| creator | Logunov, A. Malinnikova, E. Nadirashvili, N. Nazarov, F. |
| description | Let
Ω
be a bounded domain in
R
n
with
C
1
boundary and let
u
λ
be a Dirichlet Laplace eigenfunction in
Ω
with eigenvalue
λ
. We show that the
(
n
-
1
)
-dimensional Hausdorff measure of the zero set of
u
λ
does not exceed
C
(
Ω
)
λ
. This result is new even for the case of domains with
C
∞
-smooth boundary. |
| doi_str_mv | 10.1007/s00039-021-00581-5 |
| format | Article |
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Ω
be a bounded domain in
R
n
with
C
1
boundary and let
u
λ
be a Dirichlet Laplace eigenfunction in
Ω
with eigenvalue
λ
. We show that the
(
n
-
1
)
-dimensional Hausdorff measure of the zero set of
u
λ
does not exceed
C
(
Ω
)
λ
. This result is new even for the case of domains with
C
∞
-smooth boundary.</description><identifier>ISSN: 1016-443X</identifier><identifier>EISSN: 1420-8970</identifier><identifier>DOI: 10.1007/s00039-021-00581-5</identifier><language>eng</language><publisher>Cham: Springer International Publishing</publisher><subject>Analysis ; Dirichlet problem ; Domains ; Eigenvalues ; Eigenvectors ; Mathematics ; Mathematics and Statistics ; Smooth boundaries ; Upper bounds</subject><ispartof>Geometric and functional analysis, 2021-10, Vol.31 (5), p.1219-1244</ispartof><rights>The Author(s), under exclusive licence to Springer Nature Switzerland AG 2021</rights><rights>The Author(s), under exclusive licence to Springer Nature Switzerland AG 2021.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c319t-3fcc96622b311a4e37b6b34127af978208dd6e56aa8820614e156790a8714a4a3</citedby><cites>FETCH-LOGICAL-c319t-3fcc96622b311a4e37b6b34127af978208dd6e56aa8820614e156790a8714a4a3</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/s00039-021-00581-5$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00039-021-00581-5$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27901,27902,41464,42533,51294</link.rule.ids></links><search><creatorcontrib>Logunov, A.</creatorcontrib><creatorcontrib>Malinnikova, E.</creatorcontrib><creatorcontrib>Nadirashvili, N.</creatorcontrib><creatorcontrib>Nazarov, F.</creatorcontrib><title>The sharp upper bound for the area of the nodal sets of Dirichlet Laplace eigenfunctions</title><title>Geometric and functional analysis</title><addtitle>Geom. Funct. Anal</addtitle><description>Let
Ω
be a bounded domain in
R
n
with
C
1
boundary and let
u
λ
be a Dirichlet Laplace eigenfunction in
Ω
with eigenvalue
λ
. We show that the
(
n
-
1
)
-dimensional Hausdorff measure of the zero set of
u
λ
does not exceed
C
(
Ω
)
λ
. This result is new even for the case of domains with
C
∞
-smooth boundary.</description><subject>Analysis</subject><subject>Dirichlet problem</subject><subject>Domains</subject><subject>Eigenvalues</subject><subject>Eigenvectors</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Smooth boundaries</subject><subject>Upper bounds</subject><issn>1016-443X</issn><issn>1420-8970</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2021</creationdate><recordtype>article</recordtype><recordid>eNp9kEtLxDAUhYMoOD7-gKuA6-jNu1nK-IQBNyPMLqRtOtOhNjVpF_57M1PBnav7-s65cBC6oXBHAfR9AgBuCDBKAGRBiTxBCyoYkMJoOM09UEWE4JtzdJHSPuNSCrlAm_XO47RzccDTMPiIyzD1NW5CxGO-uOgdDs2x70PtOpz8mA6bxza21a7zI165oXOVx77d-r6Z-mpsQ5-u0FnjuuSvf-sl-nh-Wi9fyer95W35sCIVp2YkvKkqoxRjJafUCc91qUouKNOuMbpgUNS18lI5V-RBUeGpVNqAKzQVTjh-iW5n3yGGr8mn0e7DFPv80rKMa6OM4JliM1XFkFL0jR1i--nit6VgDwnaOUGbE7THBK3MIj6LUob7rY9_1v-ofgDupnJM</recordid><startdate>20211001</startdate><enddate>20211001</enddate><creator>Logunov, A.</creator><creator>Malinnikova, E.</creator><creator>Nadirashvili, N.</creator><creator>Nazarov, F.</creator><general>Springer International Publishing</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20211001</creationdate><title>The sharp upper bound for the area of the nodal sets of Dirichlet Laplace eigenfunctions</title><author>Logunov, A. ; Malinnikova, E. ; Nadirashvili, N. ; Nazarov, F.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c319t-3fcc96622b311a4e37b6b34127af978208dd6e56aa8820614e156790a8714a4a3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2021</creationdate><topic>Analysis</topic><topic>Dirichlet problem</topic><topic>Domains</topic><topic>Eigenvalues</topic><topic>Eigenvectors</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Smooth boundaries</topic><topic>Upper bounds</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Logunov, A.</creatorcontrib><creatorcontrib>Malinnikova, E.</creatorcontrib><creatorcontrib>Nadirashvili, N.</creatorcontrib><creatorcontrib>Nazarov, F.</creatorcontrib><collection>CrossRef</collection><jtitle>Geometric and functional analysis</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Logunov, A.</au><au>Malinnikova, E.</au><au>Nadirashvili, N.</au><au>Nazarov, F.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>The sharp upper bound for the area of the nodal sets of Dirichlet Laplace eigenfunctions</atitle><jtitle>Geometric and functional analysis</jtitle><stitle>Geom. Funct. Anal</stitle><date>2021-10-01</date><risdate>2021</risdate><volume>31</volume><issue>5</issue><spage>1219</spage><epage>1244</epage><pages>1219-1244</pages><issn>1016-443X</issn><eissn>1420-8970</eissn><abstract>Let
Ω
be a bounded domain in
R
n
with
C
1
boundary and let
u
λ
be a Dirichlet Laplace eigenfunction in
Ω
with eigenvalue
λ
. We show that the
(
n
-
1
)
-dimensional Hausdorff measure of the zero set of
u
λ
does not exceed
C
(
Ω
)
λ
. This result is new even for the case of domains with
C
∞
-smooth boundary.</abstract><cop>Cham</cop><pub>Springer International Publishing</pub><doi>10.1007/s00039-021-00581-5</doi><tpages>26</tpages></addata></record> |
| fulltext | fulltext |
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| ispartof | Geometric and functional analysis, 2021-10, Vol.31 (5), p.1219-1244 |
| issn | 1016-443X 1420-8970 |
| language | eng |
| recordid | cdi_proquest_journals_2614796943 |
| source | Springer Nature - Complete Springer Journals |
| subjects | Analysis Dirichlet problem Domains Eigenvalues Eigenvectors Mathematics Mathematics and Statistics Smooth boundaries Upper bounds |
| title | The sharp upper bound for the area of the nodal sets of Dirichlet Laplace eigenfunctions |
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