# American Institute of Mathematical Sciences

## 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  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 & Continuous Dynamical Systems - B, 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.  Google Scholar [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.  Google Scholar [3] G. Alessandrini, E. 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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [7] K. Astala, T. Iwaniec and E. Saksman, Beltrami operators in the plane, Duke Math. J., 107 (2001), 27-56.  doi: 10.1215/S0012-7094-01-10713-8.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [11] T. Brander, J. 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.  Google Scholar [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.  Google Scholar [13] T. Brander, B. Harrach, M. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] J. Duoandikoetxea, Fourier Analysis, Graduate Studies in Mathematics, 29, American Mathematical Society, Providence, RI, 2001.  Google Scholar [17] E. Francini, C.-L. Lin, S. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] C.-Y. Guo, M. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] J. J. Manfredi, $p$-harmonic functions in the plane, Proc. Amer. Math. Soc., 103 (1988), 473-479.  doi: 10.2307/2047164.  Google Scholar [26] A. I. Markushevich, Theory of Functions of a Complex Variable, English edition, 3, Prentice-Hall Inc., Englewood Cliffs, NJ, 1967.  Google Scholar [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.   Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [3] G. Alessandrini, E. 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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [7] K. Astala, T. Iwaniec and E. Saksman, Beltrami operators in the plane, Duke Math. J., 107 (2001), 27-56.  doi: 10.1215/S0012-7094-01-10713-8.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [11] T. Brander, J. 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.  Google Scholar [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.  Google Scholar [13] T. Brander, B. Harrach, M. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] J. Duoandikoetxea, Fourier Analysis, Graduate Studies in Mathematics, 29, American Mathematical Society, Providence, RI, 2001.  Google Scholar [17] E. Francini, C.-L. Lin, S. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] C.-Y. Guo, M. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] J. J. Manfredi, $p$-harmonic functions in the plane, Proc. Amer. Math. Soc., 103 (1988), 473-479.  doi: 10.2307/2047164.  Google Scholar [26] A. I. Markushevich, Theory of Functions of a Complex Variable, English edition, 3, Prentice-Hall Inc., Englewood Cliffs, NJ, 1967.  Google Scholar [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.   Google Scholar [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.  Google Scholar
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