L2 -Boundedness of Gradients of Single Layer Potentials for Elliptic Operators with Coefficients of Dini Mean Oscillation-Type
We consider a uniformly elliptic operator LA in divergence form associated with an (n+ 1) × (n+ 1) -matrix A with real, merely bounded, and possibly non-symmetric coefficients. If [Equation not available: see fulltext.]then, under suitable Dini-type assumptions on ωA, we prove the following: if μ is...
| Autores: | , , , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2023 |
| País: | España |
| Institución: | Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) |
| Repositorio: | Recercat. Dipósit de la Recerca de Catalunya |
| OAI Identifier: | oai:recercat.cat:2072/535447 |
| Acceso en línea: | http://hdl.handle.net/2072/535447 |
| Access Level: | acceso abierto |
| Palabra clave: | David–Semmes problem Dini mean oscillation Layer potentials Rectifiability Riesz transform Second order elliptic equations Uniform rectifiability |
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España |
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| dc.title.none.fl_str_mv |
L2 -Boundedness of Gradients of Single Layer Potentials for Elliptic Operators with Coefficients of Dini Mean Oscillation-Type |
| title |
L2 -Boundedness of Gradients of Single Layer Potentials for Elliptic Operators with Coefficients of Dini Mean Oscillation-Type |
| spellingShingle |
L2 -Boundedness of Gradients of Single Layer Potentials for Elliptic Operators with Coefficients of Dini Mean Oscillation-Type Molero, A. David–Semmes problem Dini mean oscillation Layer potentials Rectifiability Riesz transform Second order elliptic equations Uniform rectifiability |
| title_short |
L2 -Boundedness of Gradients of Single Layer Potentials for Elliptic Operators with Coefficients of Dini Mean Oscillation-Type |
| title_full |
L2 -Boundedness of Gradients of Single Layer Potentials for Elliptic Operators with Coefficients of Dini Mean Oscillation-Type |
| title_fullStr |
L2 -Boundedness of Gradients of Single Layer Potentials for Elliptic Operators with Coefficients of Dini Mean Oscillation-Type |
| title_full_unstemmed |
L2 -Boundedness of Gradients of Single Layer Potentials for Elliptic Operators with Coefficients of Dini Mean Oscillation-Type |
| title_sort |
L2 -Boundedness of Gradients of Single Layer Potentials for Elliptic Operators with Coefficients of Dini Mean Oscillation-Type |
| dc.creator.none.fl_str_mv |
Molero, A. Mourgoglou, M. Puliatti, C. Tolsa, X. |
| author |
Molero, A. |
| author_facet |
Molero, A. Mourgoglou, M. Puliatti, C. Tolsa, X. |
| author_role |
author |
| author2 |
Mourgoglou, M. Puliatti, C. Tolsa, X. |
| author2_role |
author author author |
| dc.subject.none.fl_str_mv |
David–Semmes problem Dini mean oscillation Layer potentials Rectifiability Riesz transform Second order elliptic equations Uniform rectifiability |
| topic |
David–Semmes problem Dini mean oscillation Layer potentials Rectifiability Riesz transform Second order elliptic equations Uniform rectifiability |
| description |
We consider a uniformly elliptic operator LA in divergence form associated with an (n+ 1) × (n+ 1) -matrix A with real, merely bounded, and possibly non-symmetric coefficients. If [Equation not available: see fulltext.]then, under suitable Dini-type assumptions on ωA, we prove the following: if μ is a compactly supported Radon measure in Rn+1, n≥ 2 , and Tμf(x)=∫∇xΓA(x,y)f(y)dμ(y) denotes the gradient of the single layer potential associated with LA, then 1+‖Tμ‖L2(μ)→L2(μ)≈1+‖Rμ‖L2(μ)→L2(μ),where Rμ indicates the n-dimensional Riesz transform. This allows us to provide a direct generalization of some deep geometric results, initially obtained for Rμ, which were recently extended to Tμ associated with LA with Hölder continuous coefficients. In particular, we show the following: (1)If μ is an n-Ahlfors-David-regular measure on Rn+1 with compact support, then Tμ is bounded on L2(μ) if and only if μ is uniformly n-rectifiable.(2)Let E⊂ Rn+1 be compact and Hn(E) < ∞. If THn|E is bounded on L2(Hn| E) , then E is n-rectifiable.(3)If μ is a non-zero measure on Rn+1 such that lim supr→0μ(B(x,r))(2r)n is positive and finite for μ-a.e. x∈ Rn+1 and lim infr→0μ(B(x,r))(2r)n vanishes for μ-a.e. x∈ Rn+1, then the operator Tμ is not bounded on L2(μ).(4)Finally, we prove that if μ is a Radon measure on Rn+1 with compact support which satisfies a proper set of local conditions at the level of a ball B= B(x, r) ⊂ Rn+1 such that μ(B) ≈ rn and r is small enough, then a significant portion of the support of μ| B can be covered by a uniformly n-rectifiable set. These assumptions include a flatness condition, the L2(μ) -boundedness of Tμ on a large enough dilation of B, and the smallness of the mean oscillation of Tμ at the level of B. © 2023, The Author(s). |
| publishDate |
2023 |
| dc.date.none.fl_str_mv |
2023 |
| dc.type.none.fl_str_mv |
info:eu-repo/semantics/article info:eu-repo/semantics/publishedVersion |
| format |
article |
| status_str |
publishedVersion |
| dc.identifier.none.fl_str_mv |
http://hdl.handle.net/2072/535447 |
| url |
http://hdl.handle.net/2072/535447 |
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Inglés |
| language_invalid_str_mv |
Inglés |
| dc.relation.none.fl_str_mv |
Archive for Rational Mechanics and Analysis |
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info:eu-repo/semantics/openAccess |
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openAccess |
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59 p. application/pdf |
| dc.publisher.none.fl_str_mv |
Springer Science and Business Media Deutschland GmbH |
| publisher.none.fl_str_mv |
Springer Science and Business Media Deutschland GmbH |
| dc.source.none.fl_str_mv |
RECERCAT (Dipòsit de la Recerca de Catalunya) reponame:Recercat. Dipósit de la Recerca de Catalunya instname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) |
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Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) |
| reponame_str |
Recercat. Dipósit de la Recerca de Catalunya |
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Recercat. Dipósit de la Recerca de Catalunya |
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1869422621650583552 |
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L2 -Boundedness of Gradients of Single Layer Potentials for Elliptic Operators with Coefficients of Dini Mean Oscillation-TypeMolero, A.Mourgoglou, M.Puliatti, C.Tolsa, X.David–Semmes problemDini mean oscillationLayer potentialsRectifiabilityRiesz transformSecond order elliptic equationsUniform rectifiabilityWe consider a uniformly elliptic operator LA in divergence form associated with an (n+ 1) × (n+ 1) -matrix A with real, merely bounded, and possibly non-symmetric coefficients. If [Equation not available: see fulltext.]then, under suitable Dini-type assumptions on ωA, we prove the following: if μ is a compactly supported Radon measure in Rn+1, n≥ 2 , and Tμf(x)=∫∇xΓA(x,y)f(y)dμ(y) denotes the gradient of the single layer potential associated with LA, then 1+‖Tμ‖L2(μ)→L2(μ)≈1+‖Rμ‖L2(μ)→L2(μ),where Rμ indicates the n-dimensional Riesz transform. This allows us to provide a direct generalization of some deep geometric results, initially obtained for Rμ, which were recently extended to Tμ associated with LA with Hölder continuous coefficients. In particular, we show the following: (1)If μ is an n-Ahlfors-David-regular measure on Rn+1 with compact support, then Tμ is bounded on L2(μ) if and only if μ is uniformly n-rectifiable.(2)Let E⊂ Rn+1 be compact and Hn(E) < ∞. If THn|E is bounded on L2(Hn| E) , then E is n-rectifiable.(3)If μ is a non-zero measure on Rn+1 such that lim supr→0μ(B(x,r))(2r)n is positive and finite for μ-a.e. x∈ Rn+1 and lim infr→0μ(B(x,r))(2r)n vanishes for μ-a.e. x∈ Rn+1, then the operator Tμ is not bounded on L2(μ).(4)Finally, we prove that if μ is a Radon measure on Rn+1 with compact support which satisfies a proper set of local conditions at the level of a ball B= B(x, r) ⊂ Rn+1 such that μ(B) ≈ rn and r is small enough, then a significant portion of the support of μ| B can be covered by a uniformly n-rectifiable set. These assumptions include a flatness condition, the L2(μ) -boundedness of Tμ on a large enough dilation of B, and the smallness of the mean oscillation of Tμ at the level of B. © 2023, The Author(s).Horizon 2020 Framework Programme, H2020: PID2020-114167GB-I00; H2020 European Research Council, ERC; es:CEI; fr:CER; pl:ERBN: 101018680, CEX2020-001084-M; Ministerio de Ciencia, Innovación y Universidades, MCIU; Hezkuntza, Hizkuntza Politika Eta Kultura Saila, Eusko Jaurlaritza; European Research Council, ERC; Deutsche Forschungsgemeinschaft, DFG: Strategy-EXC-2047/1-90685813; Eusko Jaurlaritza: PGC2018-094522-B-I00; Ministerio de Economía y Competitividad, MINECO: IT-1247-19, PID2020-118986GB-I00; Ministerio de Ciencia e Innovación, MICINN; Ekonomiaren Garapen eta Lehiakortasun Saila, Eusko Jaurlaritza; Ministerio de Asuntos Económicos y Transformación Digital, Gobierno de España, MINECO: BES-2017-081272, MTM-2016-77635-P. A.M. was supported by the predoctoral grant BES-2017-081272 and was partially supported by the grant MTM-2016-77635-P of the Ministerio de Economía y Competitividad (Spain). M.M. was supported by IKERBASQUE and partially supported by the grant PID2020-118986GB-I00 of the Ministerio de Economía y Competitividad (Spain), and by IT-1247-19 (Basque Government). C.P. was supported by the grant IT-1247-19 (Basque Government) and partially supported by PID2020-118986GB-I00 (Ministerio de Economía y Competitividad, Spain) and PGC2018-094522-B-I00 (Ministerio de Ciencia e Innovación, Spain). X.T. is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement 101018680) and María de Maeztu Program for Centers and Units of Excellence (CEX2020-001084-M). He is also partially supported by the grant PID2020-114167GB-I00 of the Ministerio de Economía y Competitividad (Spain). This material is partially based upon work funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy-EXC-2047/1-90685813. while M.M., C.P., and X.T. were in residence at the Hausdorff Research Institute in Spring 2022 during the program “Interactions between geometric measure theory, singular integrals, and PDEs..Springer Science and Business Media Deutschland GmbH2023info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersion59 p.application/pdfhttp://hdl.handle.net/2072/535447RECERCAT (Dipòsit de la Recerca de Catalunya)reponame:Recercat. Dipósit de la Recerca de Catalunyainstname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)InglésArchive for Rational Mechanics and AnalysisL'accés als continguts d'aquest document queda condicionat a l'acceptació de les condicions d'ús establertes per la següent llicència Creative Commons: https://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccessoai:recercat.cat:2072/5354472026-05-29T05:05:01Z |
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