Perturbations of elliptic operators in 1-sided chord-arc domains. Part I: Small and large perturbation for symmetric operators
Let $\Omega\subset\re^{n+1}$, $n\ge 2$, be a 1-sided chord-arc domain, that is, a domain which satisfies interior Corkscrew and Harnack Chain conditions (these are respectively scale-invariant/quantitative versions of the openness and path-connectedness), and whose boundary $\partial\Omega$ is $n$-d...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Estado: | Versión aceptada para publicación |
| Fecha de publicación: | 2019 |
| País: | España |
| Institución: | Consejo Superior de Investigaciones Científicas (CSIC) |
| Repositorio: | DIGITAL.CSIC. Repositorio Institucional del CSIC |
| OAI Identifier: | oai:digital.csic.es:10261/185678 |
| Acceso en línea: | http://hdl.handle.net/10261/185678 |
| Access Level: | acceso abierto |
| Palabra clave: | Elliptic measure Poisson kernel Carleson measures Muckenhoupt weights |
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Perturbations of elliptic operators in 1-sided chord-arc domains. Part I: Small and large perturbation for symmetric operatorsCavero, JuanHofmann, SteveMartell, José MaríaElliptic measurePoisson kernelCarleson measuresMuckenhoupt weightsLet $\Omega\subset\re^{n+1}$, $n\ge 2$, be a 1-sided chord-arc domain, that is, a domain which satisfies interior Corkscrew and Harnack Chain conditions (these are respectively scale-invariant/quantitative versions of the openness and path-connectedness), and whose boundary $\partial\Omega$ is $n$-dimensional Ahlfors regular. Consider $L_0$ and $L$ two real symmetric divergence form elliptic operators and let $\omega_{L_0}$, $\omega_L$ be the associated elliptic measures. We show that if $\omega_{L_0}\in A_\infty(\sigma)$, where $\sigma=H^n\rest{\partial\Omega}$, and $L$ is a perturbation of $L_0$ (in the sense that the discrepancy between $L_0$ and $L$ satisfies certain Carleson measure condition), then $\omega_L\in A_\infty(\sigma)$. Moreover, if $L$ is a sufficiently small perturbation of $L_0$, then one can preserve the reverse Hölder classes, that is, if for some $1<p<\infty$, one has $\omega_{L_0}\in RH_p(\sigma)$ then $\omega_{L}\in RH_p(\sigma)$. Equivalently, if the Dirichlet problem with data in $L^{p'}(\sigma)$ is solvable for $L_0$ then so it is for $L$. These results can be seen as extensions of the perturbation theorems obtained by Dahlberg, Fefferman-Kenig-Pipher, and Milakis-Pipher-Toro in more benign settings. As a consequence of our methods we can show that for any perturbation of the Laplacian (or, more in general, of any elliptic symmetric operator with Lipschitz coefficients satisfying certain Carleson condition) if its elliptic measure belongs to $A_\infty(\sigma)$ then necessarily $\Omega$ is in fact an NTA domain (and hence chord-arc) and therefore its boundary is uniformly rectifiable.The first and third authors acknowledge financial support from the Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa” Programme for Centres of Excellence in R\&D” (SEV-2015-0554). They also acknowledge that the research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013)/ ERC agreement no. 615112 HAPDEGMT. The second author was supported by NSF grant DMS-1664047.Peer reviewedAmerican Mathematical SocietyEuropean Research CouncilMinisterio de Economía y Competitividad (España)Simons FoundationMartell, José María [0000-0001-6788-4769]Consejo Superior de Investigaciones Científicas [https://ror.org/02gfc7t72]201920192019info:eu-repo/semantics/articlehttp://purl.org/coar/resource_type/c_6501Postprintinfo:eu-repo/semantics/acceptedVersionhttp://hdl.handle.net/10261/185678reponame:DIGITAL.CSIC. Repositorio Institucional del CSICinstname:Consejo Superior de Investigaciones Científicas (CSIC)Inglés#PLACEHOLDER_PARENT_METADATA_VALUE##PLACEHOLDER_PARENT_METADATA_VALUE#info:eu-repo/grantAgreement/EC/FP7/615112info:eu-repo/grantAgreement/MINECO/Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016/SEV-2015-0554https://doi.org/10.1090/tran/7536Síinfo:eu-repo/semantics/openAccessoai:digital.csic.es:10261/1856782026-05-22T06:33:51Z |
| dc.title.none.fl_str_mv |
Perturbations of elliptic operators in 1-sided chord-arc domains. Part I: Small and large perturbation for symmetric operators |
| title |
Perturbations of elliptic operators in 1-sided chord-arc domains. Part I: Small and large perturbation for symmetric operators |
| spellingShingle |
Perturbations of elliptic operators in 1-sided chord-arc domains. Part I: Small and large perturbation for symmetric operators Cavero, Juan Elliptic measure Poisson kernel Carleson measures Muckenhoupt weights |
| title_short |
Perturbations of elliptic operators in 1-sided chord-arc domains. Part I: Small and large perturbation for symmetric operators |
| title_full |
Perturbations of elliptic operators in 1-sided chord-arc domains. Part I: Small and large perturbation for symmetric operators |
| title_fullStr |
Perturbations of elliptic operators in 1-sided chord-arc domains. Part I: Small and large perturbation for symmetric operators |
| title_full_unstemmed |
Perturbations of elliptic operators in 1-sided chord-arc domains. Part I: Small and large perturbation for symmetric operators |
| title_sort |
Perturbations of elliptic operators in 1-sided chord-arc domains. Part I: Small and large perturbation for symmetric operators |
| dc.creator.none.fl_str_mv |
Cavero, Juan Hofmann, Steve Martell, José María |
| author |
Cavero, Juan |
| author_facet |
Cavero, Juan Hofmann, Steve Martell, José María |
| author_role |
author |
| author2 |
Hofmann, Steve Martell, José María |
| author2_role |
author author |
| dc.contributor.none.fl_str_mv |
European Research Council Ministerio de Economía y Competitividad (España) Simons Foundation Martell, José María [0000-0001-6788-4769] Consejo Superior de Investigaciones Científicas [https://ror.org/02gfc7t72] |
| dc.subject.none.fl_str_mv |
Elliptic measure Poisson kernel Carleson measures Muckenhoupt weights |
| topic |
Elliptic measure Poisson kernel Carleson measures Muckenhoupt weights |
| description |
Let $\Omega\subset\re^{n+1}$, $n\ge 2$, be a 1-sided chord-arc domain, that is, a domain which satisfies interior Corkscrew and Harnack Chain conditions (these are respectively scale-invariant/quantitative versions of the openness and path-connectedness), and whose boundary $\partial\Omega$ is $n$-dimensional Ahlfors regular. Consider $L_0$ and $L$ two real symmetric divergence form elliptic operators and let $\omega_{L_0}$, $\omega_L$ be the associated elliptic measures. We show that if $\omega_{L_0}\in A_\infty(\sigma)$, where $\sigma=H^n\rest{\partial\Omega}$, and $L$ is a perturbation of $L_0$ (in the sense that the discrepancy between $L_0$ and $L$ satisfies certain Carleson measure condition), then $\omega_L\in A_\infty(\sigma)$. Moreover, if $L$ is a sufficiently small perturbation of $L_0$, then one can preserve the reverse Hölder classes, that is, if for some $1<p<\infty$, one has $\omega_{L_0}\in RH_p(\sigma)$ then $\omega_{L}\in RH_p(\sigma)$. Equivalently, if the Dirichlet problem with data in $L^{p'}(\sigma)$ is solvable for $L_0$ then so it is for $L$. These results can be seen as extensions of the perturbation theorems obtained by Dahlberg, Fefferman-Kenig-Pipher, and Milakis-Pipher-Toro in more benign settings. As a consequence of our methods we can show that for any perturbation of the Laplacian (or, more in general, of any elliptic symmetric operator with Lipschitz coefficients satisfying certain Carleson condition) if its elliptic measure belongs to $A_\infty(\sigma)$ then necessarily $\Omega$ is in fact an NTA domain (and hence chord-arc) and therefore its boundary is uniformly rectifiable. |
| publishDate |
2019 |
| dc.date.none.fl_str_mv |
2019 2019 2019 |
| dc.type.none.fl_str_mv |
info:eu-repo/semantics/article http://purl.org/coar/resource_type/c_6501 Postprint info:eu-repo/semantics/acceptedVersion |
| format |
article |
| status_str |
acceptedVersion |
| dc.identifier.none.fl_str_mv |
http://hdl.handle.net/10261/185678 |
| url |
http://hdl.handle.net/10261/185678 |
| dc.language.none.fl_str_mv |
Inglés |
| language_invalid_str_mv |
Inglés |
| dc.relation.none.fl_str_mv |
#PLACEHOLDER_PARENT_METADATA_VALUE# #PLACEHOLDER_PARENT_METADATA_VALUE# info:eu-repo/grantAgreement/EC/FP7/615112 info:eu-repo/grantAgreement/MINECO/Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016/SEV-2015-0554 https://doi.org/10.1090/tran/7536 Sí |
| dc.rights.none.fl_str_mv |
info:eu-repo/semantics/openAccess |
| eu_rights_str_mv |
openAccess |
| dc.publisher.none.fl_str_mv |
American Mathematical Society |
| publisher.none.fl_str_mv |
American Mathematical Society |
| dc.source.none.fl_str_mv |
reponame:DIGITAL.CSIC. Repositorio Institucional del CSIC instname:Consejo Superior de Investigaciones Científicas (CSIC) |
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Consejo Superior de Investigaciones Científicas (CSIC) |
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DIGITAL.CSIC. Repositorio Institucional del CSIC |
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DIGITAL.CSIC. Repositorio Institucional del CSIC |
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