An Inverse Problem for a Semilinear Elliptic Equation on Conformally Transversally Anisotropic Manifolds
Given a conformally transversally anisotropic manifold ( M , g ), we consider the semilinear elliptic equation ( - Δ g + V ) u + q u 2 = 0 on M . We show that an a priori unknown smooth function q can be uniquely determined from the knowledge of the Dirichlet-to-Neumann map associated to the equati...
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| Veröffentlicht in: | Annals of PDE 2023-12, Vol.9 (2), p.12, Article 12 |
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| creator | Feizmohammadi, Ali Liimatainen, Tony Lin, Yi-Hsuan |
| description | Given a conformally transversally anisotropic manifold (
M
,
g
), we consider the semilinear elliptic equation
(
-
Δ
g
+
V
)
u
+
q
u
2
=
0
on
M
.
We show that an a priori unknown smooth function
q
can be uniquely determined from the knowledge of the Dirichlet-to-Neumann map associated to the equation. This extends the previously known results of the works Feizmohammadi and Oksanen (J Differ Equ 269(6):4683–4719, 2020), Lassas et al. (J Math Pures Appl 145:44–82, 2021). Our proof is based on over-differentiating the equation: We linearize the equation to orders higher than the order two of the nonlinearity
q
u
2
, and introduce non-vanishing boundary traces for the linearizations. We study interactions of two or more products of the so-called Gaussian quasimode solutions to the linearized equation. We develop an asymptotic calculus to solve Laplace equations, which have these interactions as source terms. |
| doi_str_mv | 10.1007/s40818-023-00153-w |
| format | Article |
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M
,
g
), we consider the semilinear elliptic equation
(
-
Δ
g
+
V
)
u
+
q
u
2
=
0
on
M
.
We show that an a priori unknown smooth function
q
can be uniquely determined from the knowledge of the Dirichlet-to-Neumann map associated to the equation. This extends the previously known results of the works Feizmohammadi and Oksanen (J Differ Equ 269(6):4683–4719, 2020), Lassas et al. (J Math Pures Appl 145:44–82, 2021). Our proof is based on over-differentiating the equation: We linearize the equation to orders higher than the order two of the nonlinearity
q
u
2
, and introduce non-vanishing boundary traces for the linearizations. We study interactions of two or more products of the so-called Gaussian quasimode solutions to the linearized equation. We develop an asymptotic calculus to solve Laplace equations, which have these interactions as source terms.</description><identifier>ISSN: 2524-5317</identifier><identifier>EISSN: 2199-2576</identifier><identifier>DOI: 10.1007/s40818-023-00153-w</identifier><language>eng</language><publisher>Cham: Springer International Publishing</publisher><subject>Dirichlet problem ; Eigenvalues ; Euclidean space ; Inverse problems ; Laplace equation ; Manifolds ; Mathematical Methods in Physics ; Nonlinear equations ; Partial Differential Equations ; Physics ; Physics and Astronomy</subject><ispartof>Annals of PDE, 2023-12, Vol.9 (2), p.12, Article 12</ispartof><rights>The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c319t-61483c46d402304c1cffda9c024f9f2a868f81af68a54a72ec787d0d7c05fd2a3</citedby><cites>FETCH-LOGICAL-c319t-61483c46d402304c1cffda9c024f9f2a868f81af68a54a72ec787d0d7c05fd2a3</cites><orcidid>0000-0002-3850-8091</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s40818-023-00153-w$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://www.proquest.com/docview/2922078214?pq-origsite=primo$$EHTML$$P50$$Gproquest$$H</linktohtml><link.rule.ids>314,780,784,21387,21388,21389,21390,23255,27923,27924,33529,33702,33743,34004,34313,41487,42556,43658,43786,43804,43952,44066,51318,64384,64388,72240</link.rule.ids></links><search><creatorcontrib>Feizmohammadi, Ali</creatorcontrib><creatorcontrib>Liimatainen, Tony</creatorcontrib><creatorcontrib>Lin, Yi-Hsuan</creatorcontrib><title>An Inverse Problem for a Semilinear Elliptic Equation on Conformally Transversally Anisotropic Manifolds</title><title>Annals of PDE</title><addtitle>Ann. PDE</addtitle><description>Given a conformally transversally anisotropic manifold (
M
,
g
), we consider the semilinear elliptic equation
(
-
Δ
g
+
V
)
u
+
q
u
2
=
0
on
M
.
We show that an a priori unknown smooth function
q
can be uniquely determined from the knowledge of the Dirichlet-to-Neumann map associated to the equation. This extends the previously known results of the works Feizmohammadi and Oksanen (J Differ Equ 269(6):4683–4719, 2020), Lassas et al. (J Math Pures Appl 145:44–82, 2021). Our proof is based on over-differentiating the equation: We linearize the equation to orders higher than the order two of the nonlinearity
q
u
2
, and introduce non-vanishing boundary traces for the linearizations. We study interactions of two or more products of the so-called Gaussian quasimode solutions to the linearized equation. We develop an asymptotic calculus to solve Laplace equations, which have these interactions as source terms.</description><subject>Dirichlet problem</subject><subject>Eigenvalues</subject><subject>Euclidean space</subject><subject>Inverse problems</subject><subject>Laplace equation</subject><subject>Manifolds</subject><subject>Mathematical Methods in Physics</subject><subject>Nonlinear equations</subject><subject>Partial Differential Equations</subject><subject>Physics</subject><subject>Physics and Astronomy</subject><issn>2524-5317</issn><issn>2199-2576</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2023</creationdate><recordtype>article</recordtype><sourceid>ABUWG</sourceid><sourceid>AFKRA</sourceid><sourceid>AZQEC</sourceid><sourceid>BENPR</sourceid><sourceid>CCPQU</sourceid><sourceid>DWQXO</sourceid><sourceid>GNUQQ</sourceid><recordid>eNp9UMFKAzEQDaJg0f6Ap4Dn6CSb3c0eS6laqChYzyFmE92ym2yTraV_b9oVvAkDMwPvvZn3ELqhcEcByvvIQVBBgGUEgOYZ2Z-hCaNVRVheFudpzhkneUbLSzSNcQMAjHKeQzFBXzOHl-7bhGjwa_Afremw9QEr_Ga6pm2cUQEv2rbph0bjxXanhsY7nGruXQJ2qm0PeB2Ui0eR0zZzTfRD8H1iPCvXWN_W8RpdWNVGM_3tV-j9YbGeP5HVy-NyPlsRndFqIAXlItO8qHmyA1xTbW2tKg2M28oyJQphBVW2ECrnqmRGl6KsoS415LZmKrtCt6NuH_x2Z-IgN34XXDopWcUYlCJ5Tyg2onTwMQZjZR-aToWDpCCPocoxVJm-kKdQ5T6RspEUE9h9mvAn_Q_rB7iqe5U</recordid><startdate>20231201</startdate><enddate>20231201</enddate><creator>Feizmohammadi, Ali</creator><creator>Liimatainen, Tony</creator><creator>Lin, Yi-Hsuan</creator><general>Springer International Publishing</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><scope>3V.</scope><scope>7XB</scope><scope>88I</scope><scope>8FE</scope><scope>8FG</scope><scope>8FK</scope><scope>ABJCF</scope><scope>ABUWG</scope><scope>AFKRA</scope><scope>ARAPS</scope><scope>AZQEC</scope><scope>BENPR</scope><scope>BGLVJ</scope><scope>BHPHI</scope><scope>BKSAR</scope><scope>CCPQU</scope><scope>DWQXO</scope><scope>GNUQQ</scope><scope>HCIFZ</scope><scope>JQ2</scope><scope>K7-</scope><scope>L6V</scope><scope>M2P</scope><scope>M7S</scope><scope>P62</scope><scope>PCBAR</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PTHSS</scope><scope>Q9U</scope><orcidid>https://orcid.org/0000-0002-3850-8091</orcidid></search><sort><creationdate>20231201</creationdate><title>An Inverse Problem for a Semilinear Elliptic Equation on Conformally Transversally Anisotropic Manifolds</title><author>Feizmohammadi, Ali ; Liimatainen, Tony ; Lin, Yi-Hsuan</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c319t-61483c46d402304c1cffda9c024f9f2a868f81af68a54a72ec787d0d7c05fd2a3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2023</creationdate><topic>Dirichlet problem</topic><topic>Eigenvalues</topic><topic>Euclidean space</topic><topic>Inverse problems</topic><topic>Laplace equation</topic><topic>Manifolds</topic><topic>Mathematical Methods in Physics</topic><topic>Nonlinear equations</topic><topic>Partial Differential Equations</topic><topic>Physics</topic><topic>Physics and Astronomy</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Feizmohammadi, Ali</creatorcontrib><creatorcontrib>Liimatainen, Tony</creatorcontrib><creatorcontrib>Lin, Yi-Hsuan</creatorcontrib><collection>CrossRef</collection><collection>ProQuest Central (Corporate)</collection><collection>ProQuest Central (purchase pre-March 2016)</collection><collection>Science Database (Alumni Edition)</collection><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>ProQuest Central (Alumni) (purchase pre-March 2016)</collection><collection>Materials Science & Engineering Collection</collection><collection>ProQuest Central (Alumni Edition)</collection><collection>ProQuest Central UK/Ireland</collection><collection>Advanced Technologies & Aerospace Collection</collection><collection>ProQuest Central Essentials</collection><collection>ProQuest Central</collection><collection>Technology Collection</collection><collection>Natural Science Collection</collection><collection>Earth, Atmospheric & Aquatic Science Collection</collection><collection>ProQuest One Community College</collection><collection>ProQuest Central Korea</collection><collection>ProQuest Central Student</collection><collection>SciTech Premium Collection</collection><collection>ProQuest Computer Science Collection</collection><collection>Computer Science Database</collection><collection>ProQuest Engineering Collection</collection><collection>ProQuest Science Journals</collection><collection>Engineering Database</collection><collection>ProQuest Advanced Technologies & Aerospace Collection</collection><collection>Earth, Atmospheric & Aquatic Science Database</collection><collection>ProQuest One Academic Eastern Edition (DO NOT USE)</collection><collection>ProQuest One Academic</collection><collection>ProQuest One Academic UKI Edition</collection><collection>Engineering Collection</collection><collection>ProQuest Central Basic</collection><jtitle>Annals of PDE</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Feizmohammadi, Ali</au><au>Liimatainen, Tony</au><au>Lin, Yi-Hsuan</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>An Inverse Problem for a Semilinear Elliptic Equation on Conformally Transversally Anisotropic Manifolds</atitle><jtitle>Annals of PDE</jtitle><stitle>Ann. PDE</stitle><date>2023-12-01</date><risdate>2023</risdate><volume>9</volume><issue>2</issue><spage>12</spage><pages>12-</pages><artnum>12</artnum><issn>2524-5317</issn><eissn>2199-2576</eissn><abstract>Given a conformally transversally anisotropic manifold (
M
,
g
), we consider the semilinear elliptic equation
(
-
Δ
g
+
V
)
u
+
q
u
2
=
0
on
M
.
We show that an a priori unknown smooth function
q
can be uniquely determined from the knowledge of the Dirichlet-to-Neumann map associated to the equation. This extends the previously known results of the works Feizmohammadi and Oksanen (J Differ Equ 269(6):4683–4719, 2020), Lassas et al. (J Math Pures Appl 145:44–82, 2021). Our proof is based on over-differentiating the equation: We linearize the equation to orders higher than the order two of the nonlinearity
q
u
2
, and introduce non-vanishing boundary traces for the linearizations. We study interactions of two or more products of the so-called Gaussian quasimode solutions to the linearized equation. We develop an asymptotic calculus to solve Laplace equations, which have these interactions as source terms.</abstract><cop>Cham</cop><pub>Springer International Publishing</pub><doi>10.1007/s40818-023-00153-w</doi><orcidid>https://orcid.org/0000-0002-3850-8091</orcidid></addata></record> |
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| subjects | Dirichlet problem Eigenvalues Euclidean space Inverse problems Laplace equation Manifolds Mathematical Methods in Physics Nonlinear equations Partial Differential Equations Physics Physics and Astronomy |
| title | An Inverse Problem for a Semilinear Elliptic Equation on Conformally Transversally Anisotropic Manifolds |
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