April  2021, 26(4): 2011-2024. doi: 10.3934/dcdsb.2020188

Size estimates for the weighted p-Laplace equation with one measurement

1. 

Mathematics, Indian Institute of Science Education and Research, Bhopal, India

2. 

Institute of Applied Mathematical Sciences, NCTS, National Taiwan University, Taipei 106, Taiwan

Received  July 2019 Revised  March 2020 Published  April 2021 Early access  June 2020

Fund Project: Kar was supported by NCTS-National Taiwan University, Taiwan and would like to acknowledge the Initiation Grant (INST/MATH/2018043) from the Indian Institute of Science Education and Research Bhopal, India.
Wang is supported in part by MOST 105-2115-M-002-014-MY3.
This paper is dedicated to Professor Sze-Bi Hsu on the occasion of his retirement

In this work, we are concerned with the problem of estimating the size of an inclusion embedded in an object laying in the two dimensional domain. We assume that the object is occupied by an exotic material which obeys a nonlinear Ohms' law. In view of the assumption of the power law, we thus consider the weighted $ p $-Laplace equation as a model problem in this case. Using only one voltage-current measurement, we give upper and lower bounds of the size of the inclusion.

Citation: Manas Kar, Jenn-Nan Wang. Size estimates for the weighted p-Laplace equation with one measurement. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 2011-2024. doi: 10.3934/dcdsb.2020188
References:
[1]

L. Ahlfors and L. Bers, Riemann's mapping theorem for variable metrics, Ann. of Math. (2), 72 (1960), 385-404.  doi: 10.2307/1970141.

[2]

G. Alessandrini and E. Rosset, The inverse conductivity problem with one measurement: Bounds on the size of the unknown object, SIAM J. Appl. Math., 58 (1998), 1060-1071.  doi: 10.1137/S0036139996306468.

[3]

G. AlessandriniE. Rosset and J. K. Seo, Optimal size estimates for the inverse conductivity problem with one measurement, Proc. Amer. Math. Soc., 128 (2000), 53-64.  doi: 10.1090/S0002-9939-99-05474-X.

[4]

G. Alessandrini, Critical points of solutions to the $p$-Laplace equation in dimension two, Boll. Un. Mat. Ital. A (7), 1 (1987), 239-246. 

[5]

G. Alessandrini, L. Rondi, E. Rosset and S. Vessella, The stability for the Cauchy problem for elliptic equations, Inverse Problems, 25 (2009), 47 pp. doi: 10.1088/0266-5611/25/12/123004.

[6]

K. Astala, T. Iwaniec and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Princeton Mathematical Series, 48, Princeton University Press, Princeton, NJ, 2009.

[7]

K. AstalaT. Iwaniec and E. Saksman, Beltrami operators in the plane, Duke Math. J., 107 (2001), 27-56.  doi: 10.1215/S0012-7094-01-10713-8.

[8]

L. Bers, F. John and M. Schechter, Partial Differential Equations, Lectures in Applied Mathematics, 3, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1964.

[9]

B. Bojarski and T. Iwaniec, $p$-harmonic equation and quasiregular mappings, in Partial Differential Equations (Warsaw, 1984), Banach Center Publ., 19, PWN, Warsaw, 1987, 25–38.

[10]

B. V. Boyarskiĭ, Generalized solutions of a system of differential equations of first order and of elliptic type with discontinuous coefficients, Mat. Sb. N.S., 43 (1957), 451-503. 

[11]

T. BranderJ. Ilmavirta and M. Kar, Superconductive and insulating inclusions for linear and non-linear conductivity equations, Inverse Probl. Imaging, 12 (2018), 91-123.  doi: 10.3934/ipi.2018004.

[12]

T. Brander, Calderón problem for the $p$-Laplacian: First order derivative of conductivity on the boundary, Proc. Amer. Math. Soc., 144 (2016), 177-189.  doi: 10.1090/proc/12681.

[13]

T. BranderB. HarrachM. Kar and M. Salo, Monotonicity and enclosure methods for the $p$-Laplace equation, SIAM J. Appl. Math., 78 (2018), 742-758.  doi: 10.1137/17M1128599.

[14]

T. Brander, M. Kar and M. Salo, Enclosure method for the $P$-Laplace equation, Inverse Problems, 31 (2015), 16 pp. doi: 10.1088/0266-5611/31/4/045001.

[15]

R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math., 51 (1974), 241-250.  doi: 10.4064/sm-51-3-241-250.

[16]

J. Duoandikoetxea, Fourier Analysis, Graduate Studies in Mathematics, 29, American Mathematical Society, Providence, RI, 2001.

[17]

E. FranciniC.-L. LinS. Vessella and J.-N. Wang, Three-region inequalities for the second order elliptic equation with discontinuous coefficients and size estimate, J. Differential Equations, 261 (2016), 5306-5323.  doi: 10.1016/j.jde.2016.08.002.

[18]

N. Garofalo and F.-H. Lin, Monotonicity properties of variational integrals, $A_p$ weights and unique continuation, Indiana Univ. Math. J., 35 (1986), 245-268.  doi: 10.1512/iumj.1986.35.35015.

[19]

S. Granlund and N. Marola, On the problem of unique continuation for the $p$-Laplace equation, Nonlinear Anal., 101 (2014), 89-97.  doi: 10.1016/j.na.2014.01.020.

[20]

C.-Y. Guo and M. Kar, Quantitative uniqueness estimates for $p$-Laplace type equations in the plane, Nonlinear Anal., 143 (2016), 19-44.  doi: 10.1016/j.na.2016.04.015.

[21]

C.-Y. GuoM. Kar and M. Salo, Inverse problems for $p$-Laplace type equations under monotonicity assumptions, Rend. Istit. Mat. Univ. Trieste, 48 (2016), 79-99.  doi: 10.13137/2464-8728/13152.

[22]

J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1993.

[23]

G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219.  doi: 10.1016/0362-546X(88)90053-3.

[24]

P. Lindqvist, Notes on the $p$-Laplace Equation, Reports of University of Jyväskylä Department of Mathematics and Statistics, 102, University of Jyväskylä, Jyväskylä, Finland, 2006.

[25]

J. J. Manfredi, $p$-harmonic functions in the plane, Proc. Amer. Math. Soc., 103 (1988), 473-479.  doi: 10.2307/2047164.

[26]

A. I. Markushevich, Theory of Functions of a Complex Variable, English edition, 3, Prentice-Hall Inc., Englewood Cliffs, NJ, 1967.

[27]

T. Nguyen and J.-N. Wang, Estimate of an inclusion in a body with discontinuous conductivity, Bull. Inst. Math. Acad. Sin. (N.S.), 9 (2014), 45-56. 

[28]

M. Salo and X. Zhong, An inverse problem for the $p$-Laplacian: Boundary determination, SIAM J. Math. Anal., 44 (2012), 2474–2495. doi: 10.1137/110838224.

show all references

References:
[1]

L. Ahlfors and L. Bers, Riemann's mapping theorem for variable metrics, Ann. of Math. (2), 72 (1960), 385-404.  doi: 10.2307/1970141.

[2]

G. Alessandrini and E. Rosset, The inverse conductivity problem with one measurement: Bounds on the size of the unknown object, SIAM J. Appl. Math., 58 (1998), 1060-1071.  doi: 10.1137/S0036139996306468.

[3]

G. AlessandriniE. Rosset and J. K. Seo, Optimal size estimates for the inverse conductivity problem with one measurement, Proc. Amer. Math. Soc., 128 (2000), 53-64.  doi: 10.1090/S0002-9939-99-05474-X.

[4]

G. Alessandrini, Critical points of solutions to the $p$-Laplace equation in dimension two, Boll. Un. Mat. Ital. A (7), 1 (1987), 239-246. 

[5]

G. Alessandrini, L. Rondi, E. Rosset and S. Vessella, The stability for the Cauchy problem for elliptic equations, Inverse Problems, 25 (2009), 47 pp. doi: 10.1088/0266-5611/25/12/123004.

[6]

K. Astala, T. Iwaniec and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Princeton Mathematical Series, 48, Princeton University Press, Princeton, NJ, 2009.

[7]

K. AstalaT. Iwaniec and E. Saksman, Beltrami operators in the plane, Duke Math. J., 107 (2001), 27-56.  doi: 10.1215/S0012-7094-01-10713-8.

[8]

L. Bers, F. John and M. Schechter, Partial Differential Equations, Lectures in Applied Mathematics, 3, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1964.

[9]

B. Bojarski and T. Iwaniec, $p$-harmonic equation and quasiregular mappings, in Partial Differential Equations (Warsaw, 1984), Banach Center Publ., 19, PWN, Warsaw, 1987, 25–38.

[10]

B. V. Boyarskiĭ, Generalized solutions of a system of differential equations of first order and of elliptic type with discontinuous coefficients, Mat. Sb. N.S., 43 (1957), 451-503. 

[11]

T. BranderJ. Ilmavirta and M. Kar, Superconductive and insulating inclusions for linear and non-linear conductivity equations, Inverse Probl. Imaging, 12 (2018), 91-123.  doi: 10.3934/ipi.2018004.

[12]

T. Brander, Calderón problem for the $p$-Laplacian: First order derivative of conductivity on the boundary, Proc. Amer. Math. Soc., 144 (2016), 177-189.  doi: 10.1090/proc/12681.

[13]

T. BranderB. HarrachM. Kar and M. Salo, Monotonicity and enclosure methods for the $p$-Laplace equation, SIAM J. Appl. Math., 78 (2018), 742-758.  doi: 10.1137/17M1128599.

[14]

T. Brander, M. Kar and M. Salo, Enclosure method for the $P$-Laplace equation, Inverse Problems, 31 (2015), 16 pp. doi: 10.1088/0266-5611/31/4/045001.

[15]

R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math., 51 (1974), 241-250.  doi: 10.4064/sm-51-3-241-250.

[16]

J. Duoandikoetxea, Fourier Analysis, Graduate Studies in Mathematics, 29, American Mathematical Society, Providence, RI, 2001.

[17]

E. FranciniC.-L. LinS. Vessella and J.-N. Wang, Three-region inequalities for the second order elliptic equation with discontinuous coefficients and size estimate, J. Differential Equations, 261 (2016), 5306-5323.  doi: 10.1016/j.jde.2016.08.002.

[18]

N. Garofalo and F.-H. Lin, Monotonicity properties of variational integrals, $A_p$ weights and unique continuation, Indiana Univ. Math. J., 35 (1986), 245-268.  doi: 10.1512/iumj.1986.35.35015.

[19]

S. Granlund and N. Marola, On the problem of unique continuation for the $p$-Laplace equation, Nonlinear Anal., 101 (2014), 89-97.  doi: 10.1016/j.na.2014.01.020.

[20]

C.-Y. Guo and M. Kar, Quantitative uniqueness estimates for $p$-Laplace type equations in the plane, Nonlinear Anal., 143 (2016), 19-44.  doi: 10.1016/j.na.2016.04.015.

[21]

C.-Y. GuoM. Kar and M. Salo, Inverse problems for $p$-Laplace type equations under monotonicity assumptions, Rend. Istit. Mat. Univ. Trieste, 48 (2016), 79-99.  doi: 10.13137/2464-8728/13152.

[22]

J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1993.

[23]

G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219.  doi: 10.1016/0362-546X(88)90053-3.

[24]

P. Lindqvist, Notes on the $p$-Laplace Equation, Reports of University of Jyväskylä Department of Mathematics and Statistics, 102, University of Jyväskylä, Jyväskylä, Finland, 2006.

[25]

J. J. Manfredi, $p$-harmonic functions in the plane, Proc. Amer. Math. Soc., 103 (1988), 473-479.  doi: 10.2307/2047164.

[26]

A. I. Markushevich, Theory of Functions of a Complex Variable, English edition, 3, Prentice-Hall Inc., Englewood Cliffs, NJ, 1967.

[27]

T. Nguyen and J.-N. Wang, Estimate of an inclusion in a body with discontinuous conductivity, Bull. Inst. Math. Acad. Sin. (N.S.), 9 (2014), 45-56. 

[28]

M. Salo and X. Zhong, An inverse problem for the $p$-Laplacian: Boundary determination, SIAM J. Math. Anal., 44 (2012), 2474–2495. doi: 10.1137/110838224.

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