Calderón problem for the $p$-Laplacian: First order derivative of conductivity on the boundary
HTML articles powered by AMS MathViewer
- by Tommi Brander PDF
- Proc. Amer. Math. Soc. 144 (2016), 177-189 Request permission
Abstract:
We recover the gradient of a scalar conductivity defined on a smooth bounded open set in $\mathbb {R}^d$ from the Dirichlet to Neumann map arising from the $p$-Laplace equation. For any boundary point we recover the gradient using Dirichlet data supported on an arbitrarily small neighbourhood of the boundary point. We use a Rellich-type identity in the proof. Our results are new when $p \neq 2$. In the $p=2$ case boundary determination plays a role in several methods for recovering the conductivity in the interior.References
- Giovanni Alessandrini, Singular solutions of elliptic equations and the determination of conductivity by boundary measurements, J. Differential Equations 84 (1990), no. 2, 252–272. MR 1047569, DOI 10.1016/0022-0396(90)90078-4
- Giovanni Alessandrini and Romina Gaburro, The local Calderòn problem and the determination at the boundary of the conductivity, Comm. Partial Differential Equations 34 (2009), no. 7-9, 918–936. MR 2560305, DOI 10.1080/03605300903017397
- H. Ammari, E. Bonnetier, Y. Capdeboscq, M. Tanter, and M. Fink, Electrical impedance tomography by elastic deformation, SIAM J. Appl. Math. 68 (2008), no. 6, 1557–1573. MR 2424952, DOI 10.1137/070686408
- Gunnar Aronsson, On $p$-harmonic functions, convex duality and an asymptotic formula for injection mould filling, European J. Appl. Math. 7 (1996), no. 5, 417–437. MR 1419641, DOI 10.1017/S0956792500002473
- G. Aronsson, L. C. Evans, and Y. Wu, Fast/slow diffusion and growing sandpiles, J. Differential Equations 131 (1996), no. 2, 304–335. MR 1419017, DOI 10.1006/jdeq.1996.0166
- Gunnar Aronsson and Ulf Janfalk, On Hele-Shaw flow of power-law fluids, European J. Appl. Math. 3 (1992), no. 4, 343–366. MR 1196816, DOI 10.1017/S0956792500000905
- Guillaume Bal, Cauchy problem for ultrasound-modulated EIT, Anal. PDE 6 (2013), no. 4, 751–775. MR 3092728, DOI 10.2140/apde.2013.6.751
- Guillaume Bal and John C. Schotland, Inverse scattering and acousto-optic imaging, Phys. Rev. Lett. 104 (2010), no. 4.
- Liliana Borcea, Electrical impedance tomography, Inverse Problems 18 (2002), no. 6, R99–R136. MR 1955896, DOI 10.1088/0266-5611/18/6/201
- Tommi Brander, Manas Kar, and Mikko Salo, Enclosure method for the $P$-Laplace equation, Inverse Problems 31 (2015), no. 4, 045001, 16. MR 3320025, DOI 10.1088/0266-5611/31/4/045001
- B. H. Brown, D. C. Barber, and A. D. Seagar, Applied potential tomography: possible clinical applications, Clin. Phys. Physiol. Meas. 6 (1985), no. 2, 109.
- R. M. Brown, Recovering the conductivity at the boundary from the Dirichlet to Neumann map: a pointwise result, J. Inverse Ill-Posed Probl. 9 (2001), no. 6, 567–574. MR 1881563, DOI 10.1515/jiip.2001.9.6.567
- Alberto-P. Calderón, On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980) Soc. Brasil. Mat., Rio de Janeiro, 1980, pp. 65–73. MR 590275
- A.-P. Calderón, Lebesgue spaces of differentiable functions and distributions, Proc. Sympos. Pure Math., Vol. IV, American Mathematical Society, Providence, R.I., 1961, pp. 33–49. MR 0143037
- Luigi D’Onofrio and Tadeusz Iwaniec, Notes on $p$-harmonic analysis, The $p$-harmonic equation and recent advances in analysis, Contemp. Math., vol. 370, Amer. Math. Soc., Providence, RI, 2005, pp. 25–49. MR 2126700, DOI 10.1090/conm/370/06828
- Andoni García and Guo Zhang, Reconstruction from boundary measurements for less regular conductivities, arXiv (2012), With an appendix by R. Brown and the authors.
- Bastian Gebauer and Otmar Scherzer, Impedance-acoustic tomography, SIAM J. Appl. Math. 69 (2008), no. 2, 565–576. MR 2465856, DOI 10.1137/080715123
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
- Boaz Haberman and Daniel Tataru, Uniqueness in Calderón’s problem with Lipschitz conductivities, Duke Math. J. 162 (2013), no. 3, 496–516. MR 3024091, DOI 10.1215/00127094-2019591
- Piotr Hajłasz, Pekka Koskela, and Heli Tuominen, Measure density and extendability of Sobolev functions, Rev. Mat. Iberoam. 24 (2008), no. 2, 645–669. MR 2459208, DOI 10.4171/RMI/551
- Daniel Hauer, The $p$-Dirichlet-to-Neumann operator with applications to elliptic and parabolic problems, (2014), Preprint, retrieved 18.9.2014 from www.maths.usyd.edu.au/u/pubs/publist/preprints/2014/hauer-9.html.
- Juha Heinonen, Tero Kilpeläinen, and Olli Martio, Nonlinear potential theory of degenerate elliptic equations, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1993. Oxford Science Publications. MR 1207810
- Nicholas Hoell, Amir Moradifam, and Adrian Nachman, Current density impedance imaging of an anisotropic conductivity in a known conformal class, SIAM J. Math. Anal. 46 (2014), no. 3, 1820–1842. MR 3206987, DOI 10.1137/130911524
- M. Joy, G. Scott, and M. Henkelman, In vivo detection of applied electric currents by magnetic resonance imaging, Magnetic Resonance Imaging 7 (1989), no. 1, 89–94.
- Hyeonbae Kang and Kihyun Yun, Boundary determination of conductivities and Riemannian metrics via local Dirichlet-to-Neumann operator, SIAM J. Math. Anal. 34 (2002), no. 3, 719–735. MR 1970890, DOI 10.1137/S0036141001395042
- Kimmo Karhunen, Electrical resistance tomography imaging of concrete, Ph.D. thesis, University of Eastern Finland, 2013.
- Sungwhan Kim, Ohin Kwon, Jin Keun Seo, and Jeong-Rock Yoon, On a nonlinear partial differential equation arising in magnetic resonance electrical impedance tomography, SIAM J. Math. Anal. 34 (2002), no. 3, 511–526. MR 1970881, DOI 10.1137/S0036141001391354
- Robert Kohn and Michael Vogelius, Determining conductivity by boundary measurements, Comm. Pure Appl. Math. 37 (1984), no. 3, 289–298. MR 739921, DOI 10.1002/cpa.3160370302
- Arjan Kuijper, p-Laplacian driven image processing, ICIP 2007. IEEE International Conference on Image Processing, vol. 5, IEEE, 2007, pp. V–257.
- Olga A. Ladyzhenskaya and Nina N. Ural’tseva, Linear and quasilinear elliptic equations, Academic Press, New York-London, 1968. Translated from the Russian by Scripta Technica, Inc; Translation editor: Leon Ehrenpreis. MR 0244627
- Gary M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), no. 11, 1203–1219. MR 969499, DOI 10.1016/0362-546X(88)90053-3
- Tony Liimatainen and Mikko Salo, $n$-harmonic coordinates and the regularity of conformal mappings, Math. Res. Lett. 21 (2014), no. 2, 341–361. MR 3247061, DOI 10.4310/MRL.2014.v21.n2.a11
- Peter Lindqvist, Notes on the $p$-Laplace equation, Report. University of Jyväskylä Department of Mathematics and Statistics, vol. 102, University of Jyväskylä, Jyväskylä, 2006. MR 2242021
- Tadeusz Iwaniec and Juan J. Manfredi, Regularity of $p$-harmonic functions on the plane, Rev. Mat. Iberoamericana 5 (1989), no. 1-2, 1–19. MR 1057335, DOI 10.4171/RMI/82
- Adrian Nachman, Alexandru Tamasan, and Alexandre Timonov, Recovering the conductivity from a single measurement of interior data, Inverse Problems 25 (2009), no. 3, 035014, 16. MR 2480184, DOI 10.1088/0266-5611/25/3/035014
- Adrian I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math. (2) 143 (1996), no. 1, 71–96. MR 1370758, DOI 10.2307/2118653
- G. Nakamura, S. Siltanen, K. Tanuma, and S. Wang, Numerical recovery of conductivity at the boundary from the localized Dirichlet to Neumann map, Computing 75 (2005), no. 2-3, 197–213. MR 2163666, DOI 10.1007/s00607-004-0095-x
- Gen Nakamura and Kazumi Tanuma, Formulas for reconstructing conductivity and its normal derivative at the boundary from the localized Dirichlet to Neumann map, Recent development in theories & numerics, World Sci. Publ., River Edge, NJ, 2003, pp. 192–201. MR 2088202, DOI 10.1142/9789812704924_{0}017
- Johann Radon, Theorie und Anwendungen der absolut additiven Mengenfunktionen, S.-B. Akad. Wiss. Wien 122 (1913), 1295–1438.
- Franz Rellich, Darstellung der Eigenwerte von $\Delta u+\lambda u=0$ durch ein Randintegral, Math. Z. 46 (1940), 635–636 (German). MR 2456, DOI 10.1007/BF01181459
- Frédéric Riesz, Sur la convergence en moyenne, I, Acta Sci. Math (Szeged) 4 (1928-1929), 58–64.
- Frédéric Riesz, Sur la convergence en moyenne, II, Acta Sci. Math (Szeged) 4 (1928–1929), 182–185.
- Mikko Salo and Xiao Zhong, An inverse problem for the $p$-Laplacian: boundary determination, SIAM J. Math. Anal. 44 (2012), no. 4, 2474–2495. MR 3023384, DOI 10.1137/110838224
- G. C. Scott, M. L. G. Joy, R .L. Armstrong, and R. M. Henkelman, Measurement of nonuniform current density by magnetic resonance, IEEE Transactions on Medical Imaging 10 (1991), no. 3, 362–374.
- Samuli Siltanen and Janne P. Tamminen, Reconstructing conductivities with boundary corrected D-bar method, J. Inverse Ill-Posed Probl. 22 (2014), no. 6, 847–870. MR 3284728, DOI 10.1515/jip-2013-0042
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- John Sylvester and Gunther Uhlmann, Inverse boundary value problems at the boundary—continuous dependence, Comm. Pure Appl. Math. 41 (1988), no. 2, 197–219. MR 924684, DOI 10.1002/cpa.3160410205
- Gunther Uhlmann, Electrical impedance tomography and Calderón’s problem, Inverse problems 25 (2009), no. 12, 123011.
- Thomas H. Wolff, Gap series constructions for the $p$-Laplacian, J. Anal. Math. 102 (2007), 371–394. Paper completed by John Garnett and Jang-Mei Wu. MR 2346563, DOI 10.1007/s11854-007-0026-9
Additional Information
- Tommi Brander
- Affiliation: Department of Mathematics and Statistics, P.O. Box 35 (MaD) FI-40014 University of Jyväskylä, Finland
- Email: tommi.o.brander@jyu.fi
- Received by editor(s): March 4, 2014
- Received by editor(s) in revised form: November 26, 2014
- Published electronically: July 24, 2015
- Communicated by: Joachim Krieger
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 177-189
- MSC (2010): Primary 35R30, 35J92
- DOI: https://doi.org/10.1090/proc/12681
- MathSciNet review: 3415587