A density property for tensor products of gradients of harmonic functions and applications
We show that linear combinations of tensor products of k gradients of harmonic functions, with k at least three, are dense in C(Ω‾), for any bounded domain Ω in dimension 3 or higher. The bulk of the argument consists in showing that any smooth compactly supported k-tensor that is L2-orthogonal to a...
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| Veröffentlicht in: | Journal of functional analysis Jg. 284; H. 2; S. 109740 |
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| Sprache: | Englisch |
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| Abstract | We show that linear combinations of tensor products of k gradients of harmonic functions, with k at least three, are dense in C(Ω‾), for any bounded domain Ω in dimension 3 or higher. The bulk of the argument consists in showing that any smooth compactly supported k-tensor that is L2-orthogonal to all such products must be zero. This is done by using a Gaussian quasi-mode based construction of harmonic functions in the orthogonality relation. We then demonstrate the usefulness of this result by using it to prove uniqueness in the inverse boundary value problem for a coupled quasilinear elliptic system. The paper ends with a discussion of the corresponding property for products of two gradients of harmonic functions, and the connection of this property with the linearized anisotropic Calderón problem. |
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| AbstractList | We show that linear combinations of tensor products of k gradients of harmonic functions, with k at least three, are dense in C(Ω‾), for any bounded domain Ω in dimension 3 or higher. The bulk of the argument consists in showing that any smooth compactly supported k-tensor that is L2-orthogonal to all such products must be zero. This is done by using a Gaussian quasi-mode based construction of harmonic functions in the orthogonality relation. We then demonstrate the usefulness of this result by using it to prove uniqueness in the inverse boundary value problem for a coupled quasilinear elliptic system. The paper ends with a discussion of the corresponding property for products of two gradients of harmonic functions, and the connection of this property with the linearized anisotropic Calderón problem. |
| ArticleNumber | 109740 |
| Author | Cârstea, Cătălin I. Feizmohammadi, Ali |
| Author_xml | – sequence: 1 givenname: Cătălin I. orcidid: 0000-0001-8644-2704 surname: Cârstea fullname: Cârstea, Cătălin I. email: catalin.carstea@gmail.com organization: School of Mathematics, Sichuan University, Chengdu, Sichuan, 610064, PR China – sequence: 2 givenname: Ali surname: Feizmohammadi fullname: Feizmohammadi, Ali email: afeizmoh@fields.utoronto.ca organization: Fields Institute for Research in Mathematical Sciences, 222 College St, Toronto, M5T 3J1, Canada |
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| Cites_doi | 10.1016/j.jde.2021.02.044 10.1081/PDE-120016164 10.1215/S0012-7094-01-10837-5 10.1088/1361-6420/abcea1 10.4171/jems/649 10.1088/0266-5611/18/4/309 10.1081/PDE-120002868 10.1088/1742-6596/12/1/015 10.1007/BF00392201 10.1016/j.aml.2019.06.009 10.1007/BF02622117 10.1081/PDE-100107813 10.2140/apde.2013.6.2003 10.4310/MRL.2020.v27.n6.a10 10.1016/j.matpur.2020.11.006 10.1090/S0002-9947-1995-1311909-1 10.1002/cpa.3160471005 10.1088/1361-6420/abced7 10.4171/rmi/1242 10.1353/ajm.1997.0027 10.1090/proc/14844 10.1016/j.jde.2020.03.037 10.1016/0022-0396(91)90051-A 10.1088/0266-5611/30/3/035009 10.2307/1971291 10.1016/j.bulsci.2008.07.001 10.4310/MRL.2009.v16.n6.a4 |
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| Keywords | Gradients of harmonic functions Quasilinear elliptic systems Inverse problems |
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| Snippet | We show that linear combinations of tensor products of k gradients of harmonic functions, with k at least three, are dense in C(Ω‾), for any bounded domain Ω... |
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| SubjectTerms | Gradients of harmonic functions Inverse problems Quasilinear elliptic systems |
| Title | A density property for tensor products of gradients of harmonic functions and applications |
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