PARTIAL DATA INVERSE PROBLEM WITH Ln/2 POTENTIALS

Francis J. Chung; Leo Tzou

doi:10.1090/btran/39

PARTIAL DATA INVERSE PROBLEM WITH L^n/2 POTENTIALS

Francis J. Chung
, Leo Tzou

Mathematics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

We construct an explicit Green’s function for the conjugated Laplacian e^−ω ^・ ^x/hΔe^−ω ^・ ^x/h, which lets us control our solutions on roughly half of the boundary. We apply the Green’s function to solve a partial data inverse problem for the Schrödinger equation with potential q ∈ L^n/ ². Separately, we also use this Green’s function to derive L^p Carleman estimates similar to the ones in Kenig-Ruiz-Sogge [Duke Math. J. 55 (1987), pp. 329-347], but for functions with support up to part of the boundary. Unlike many previous results, we did not obtain the partial data result from the boundary Carleman estimate-rather, both results stem from the same explicit construction of the Green’s function. This explicit Green’s function has potential future applications in obtaining direct numerical reconstruction algorithms for partial data Calderón problems which is presently only accessible with full data [Inverse Problems 27 (2011)].

Original language	English
Pages (from-to)	97-132
Number of pages	36
Journal	Transactions of the American Mathematical Society Series B
Volume	7
DOIs	https://doi.org/10.1090/btran/39
State	Published - 2020

Bibliographical note

Publisher Copyright:
© 2020 by the authors under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0).

Funding

Received by the editors January 30, 2019. 2010 Mathematics Subject Classification. Primary 35R30. Key words and phrases. Inverse problems, partial data, Calderón problem, Carleman estimate, Green’s function. The second author was supported by ARC DP190103302 and ARC DP190103451.

Funders	Funder number
Automotive Research and Testing Center	DP190103302
Australian Research Council	DP190103451

Keywords

Calderón problem
Carleman estimate
Green’s function
Inverse problems
Partial data

ASJC Scopus subject areas

Mathematics (miscellaneous)

Access to Document

10.1090/btran/39

Cite this

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title = "PARTIAL DATA INVERSE PROBLEM WITH Ln/2 POTENTIALS",

abstract = "We construct an explicit Green{\textquoteright}s function for the conjugated Laplacian e−ω ・ x/hΔe−ω ・ x/h, which lets us control our solutions on roughly half of the boundary. We apply the Green{\textquoteright}s function to solve a partial data inverse problem for the Schr{\"o}dinger equation with potential q ∈ Ln/ 2. Separately, we also use this Green{\textquoteright}s function to derive Lp Carleman estimates similar to the ones in Kenig-Ruiz-Sogge [Duke Math. J. 55 (1987), pp. 329-347], but for functions with support up to part of the boundary. Unlike many previous results, we did not obtain the partial data result from the boundary Carleman estimate-rather, both results stem from the same explicit construction of the Green{\textquoteright}s function. This explicit Green{\textquoteright}s function has potential future applications in obtaining direct numerical reconstruction algorithms for partial data Calder{\'o}n problems which is presently only accessible with full data [Inverse Problems 27 (2011)].",

keywords = "Calder{\'o}n problem, Carleman estimate, Green{\textquoteright}s function, Inverse problems, Partial data",

author = "Chung, \{Francis J.\} and Leo Tzou",

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doi = "10.1090/btran/39",

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AU - Chung, Francis J.

AU - Tzou, Leo

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N2 - We construct an explicit Green’s function for the conjugated Laplacian e−ω ・ x/hΔe−ω ・ x/h, which lets us control our solutions on roughly half of the boundary. We apply the Green’s function to solve a partial data inverse problem for the Schrödinger equation with potential q ∈ Ln/ 2. Separately, we also use this Green’s function to derive Lp Carleman estimates similar to the ones in Kenig-Ruiz-Sogge [Duke Math. J. 55 (1987), pp. 329-347], but for functions with support up to part of the boundary. Unlike many previous results, we did not obtain the partial data result from the boundary Carleman estimate-rather, both results stem from the same explicit construction of the Green’s function. This explicit Green’s function has potential future applications in obtaining direct numerical reconstruction algorithms for partial data Calderón problems which is presently only accessible with full data [Inverse Problems 27 (2011)].

AB - We construct an explicit Green’s function for the conjugated Laplacian e−ω ・ x/hΔe−ω ・ x/h, which lets us control our solutions on roughly half of the boundary. We apply the Green’s function to solve a partial data inverse problem for the Schrödinger equation with potential q ∈ Ln/ 2. Separately, we also use this Green’s function to derive Lp Carleman estimates similar to the ones in Kenig-Ruiz-Sogge [Duke Math. J. 55 (1987), pp. 329-347], but for functions with support up to part of the boundary. Unlike many previous results, we did not obtain the partial data result from the boundary Carleman estimate-rather, both results stem from the same explicit construction of the Green’s function. This explicit Green’s function has potential future applications in obtaining direct numerical reconstruction algorithms for partial data Calderón problems which is presently only accessible with full data [Inverse Problems 27 (2011)].

KW - Calderón problem

KW - Carleman estimate

KW - Green’s function

KW - Inverse problems

KW - Partial data

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