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
Hauptverfasser:	Cârstea, Cătălin I., Feizmohammadi, Ali
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Elsevier Inc 15.01.2023
Schlagworte:	Gradients of harmonic functions Inverse problems Quasilinear elliptic systems Gradients of harmonic functions Quasilinear elliptic systems Inverse problems
ISSN:	0022-1236, 1096-0783
Online-Zugang:	Volltext
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Zusammenfassung:	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.
ISSN:	0022-1236 1096-0783
DOI:	10.1016/j.jfa.2022.109740