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Unique Continuation for the Elasticity System and a Counterexample for Second-Order Elliptic Systems

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Harmonic Analysis, Partial Differential Equations, Complex Analysis, Banach Spaces, and Operator Theory (Volume 1)

Part of the book series: Association for Women in Mathematics Series ((AWMS,volume 4))

Abstract

In this paper we study the global unique continuation property for the elasticity system and the general second-order elliptic system in two dimensions. For the isotropic and the anisotropic systems with measurable coefficients, under certain conditions on coefficients, we show that the global unique continuation property holds. On the other hand, for the anisotropic system, if the coefficients are Lipschitz, we can prove that the global unique continuation is satisfied for a more general class of media. In addition to the positive results, we also present counterexamples to unique continuation and strong unique continuation for general second elliptic systems.

Dedicated to the memory of Cora Sadosky.

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References

  1. G. Alessandrini, A simple proof of the unique continuation property for two dimensional elliptic equations in divergence form. Quaderni Matematici II, Serie 276 (Dipartimento di Scienze Matematiche, Trieste, 1992). http://www.dmi.units.it/~alessang/unique92.pdf

  2. G. Alessandrini, Strong unique continuation for general elliptic equations in 2D. J. Math. Anal. Appl. 386, 669–676 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. G. Alessandrini, L. Escauriaza, Null-controllability of one-dimensional parabolic equations. ESAIM: Control Optim. Cal. Var. 14, 284–293 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. S. Alinhac, Non-unicité pour des opérateurs différentiels à caractéristiques complexes simples. Ann. Sci. Ecole Norm. Sup. 13, 385–393 (1980)

    MathSciNet  MATH  Google Scholar 

  5. R. Banũelos, P. Janakiraman, L p-bounds for the Beurling-Ahlfors transform. Trans. Am. Math. Soc. 360, 3603–3612 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. L. Bers, L. Nirenberg, On a representation theorem for linear elliptic systems with discontinuous coefficients and its applications, in Convegno Internazionale sullen Equazioni Lineari alle Derivate Parziali, Trieste, 1954 (Edizioni Cremonese, Roma, 1955), pp. 111–140

    Google Scholar 

  7. B.V. Bojarski, Generalized solutions of a system of differential equations of first order and of elliptic type with discontinuous coefficients. Mat. Sb. N.S. 43 (85), 451–503 (1957)

    Google Scholar 

  8. A. Coffman, Y. Pan, Smooth counterexamples to strong unique continuation for a Beltrami 375 system in \(\mathbb{C}^{2}\). Comm. PDE. 37, 2228–2244 (2012)

    Article  MATH  Google Scholar 

  9. T. Iwaniec, G.C. Verchota, A.L. Vogel, The failure of rank-one connections. Arch. Ration. Mech. Anal. 163, 125–169 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. C.L. Lin, G. Nakamura, G. Uhlmann, J.N. Wang, Quantitative strong unique continuation for the Lame system with less regular coefficients. Methods Appl. Anal. 18, 85–92 (2011)

    MathSciNet  MATH  Google Scholar 

  11. S. Müller, V. Šverák, Convex integration for Lipschitz mappings and counterexamples to regularity. Ann. Math. 157, 715–742 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. J.P. Rosay, Uniqueness in rough almost complex structures and differential inequalities. Ann. Inst. Fourier (Grenoble). 60, 2261–2273 (2010)

    Google Scholar 

  13. F. Schulz, On the unique continuation property of elliptic divergence form equations in the plane. Math. Z. 228, 201–206 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

Carlos Kenig is supported in part by NSF Grant DMS-1265249. Jenn-Nan Wang is supported in part by MOST Grant 102-2918-I-002-009 and 102-2115-M-002-009-MY3.

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Authors and Affiliations

  1. Department of Mathematics, University of Chicago, Chicago, IL, 60637, USA

    Carlos Kenig

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

    Jenn-Nan Wang

Authors
  1. Carlos Kenig
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  2. Jenn-Nan Wang
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Corresponding author

Correspondence to Jenn-Nan Wang .

Editor information

Editors and Affiliations

  1. Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico, USA

    María Cristina Pereyra

  2. Department of Mathematics, Venezuelan Institute for Scientific Research, Caracas, Venezuela

    Stefania Marcantognini

  3. Department of Mathematical Sciences, Georgia Southern University, Statesboro, Georgia, USA

    Alexander M. Stokolos

  4. Department of Mathematics and Actu rial Science, Roosevelt University, Chicago, Illinois, USA

    Wilfredo Urbina

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© 2016 Springer International Publishing Switzerland

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Kenig, C., Wang, JN. (2016). Unique Continuation for the Elasticity System and a Counterexample for Second-Order Elliptic Systems. In: Pereyra, M., Marcantognini, S., Stokolos, A., Urbina, W. (eds) Harmonic Analysis, Partial Differential Equations, Complex Analysis, Banach Spaces, and Operator Theory (Volume 1). Association for Women in Mathematics Series, vol 4. Springer, Cham. https://doi.org/10.1007/978-3-319-30961-3_10

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