• Previous Article
    Global well-posedness of non-isothermal inhomogeneous nematic liquid crystal flows
  • DCDS-B Home
  • This Issue
  • Next Article
    Global existence and Gevrey regularity to the Navier-Stokes-Nernst-Planck-Poisson system in critical Besov-Morrey spaces
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  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. 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.  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. 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.  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. 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.  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. 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.  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. 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.  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. 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.  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

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.  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. 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.  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. 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.  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. 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.  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. 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.  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. 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.  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. 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.  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

[1]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384

[2]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

[3]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276

[4]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

[5]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445

[6]

Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119

[7]

Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 657-680. doi: 10.3934/dcds.2020292

[8]

Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249

[9]

Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020377

[10]

Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168

[11]

Cheng Peng, Zhaohui Tang, Weihua Gui, Qing Chen, Jing He. A bidirectional weighted boundary distance algorithm for time series similarity computation based on optimized sliding window size. Journal of Industrial & Management Optimization, 2021, 17 (1) : 205-220. doi: 10.3934/jimo.2019107

[12]

Tien-Yu Lin, Bhaba R. Sarker, Chien-Jui Lin. An optimal setup cost reduction and lot size for economic production quantity model with imperfect quality and quantity discounts. Journal of Industrial & Management Optimization, 2021, 17 (1) : 467-484. doi: 10.3934/jimo.2020043

[13]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340

[14]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217

[15]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

[16]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442

[17]

Noriyoshi Fukaya. Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential. Communications on Pure & Applied Analysis, 2021, 20 (1) : 121-143. doi: 10.3934/cpaa.2020260

[18]

Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326

[19]

Marc Homs-Dones. A generalization of the Babbage functional equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 899-919. doi: 10.3934/dcds.2020303

[20]

Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020378

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (33)
  • HTML views (178)
  • Cited by (0)

Other articles
by authors

[Back to Top]