Advertisement

Mathematische Zeitschrift

, Volume 291, Issue 1–2, pp 227–244 | Cite as

On quantitative uniqueness for elliptic equations

  • Guher Camliyurt
  • Igor KukavicaEmail author
  • Fei Wang
Article
  • 81 Downloads

Abstract

We address the question of quantitative uniqueness for the equation \(\Delta u=V u\) with either periodic or Dirichlet boundary conditions in a disk. We construct solutions u corresponding to potential functions V such that u vanishes of order \(\mathrm{const} \Vert V\Vert _{L^\infty }^{2/3}\). The example also shows sharpness of recently obtained bounds in the case of a parabolic equation \(u_t-\Delta u= V u\).

Notes

Acknowledgements

The authors were supported in part by the NSF Grant DMS-1615239.

References

  1. 1.
    Agmon, S.: Unicité et convexité dans les problémes différentiels, Séminaire de Mathématiques Supérieures, No. 13 (Été, 1965), Les Presses de l’Université de Montréal, Montreal (1966)Google Scholar
  2. 2.
    Agmon, S., Nirenberg, L.: Lower bounds and uniqueness theorems for solutions of differential equations in a Hilbert space. Commun. Pure Appl. Math. 20, 207–229 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alessandrini, G., Escauriaza, L.: Null-controllability of one-dimensional parabolic equations. ESAIM Control Optim. Calc. Var. 14(2), 284–293 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Alessandrini, G., Morassi, A., Rosset, E., Vessella, S.: On doubling inequalities for elliptic systems. J. Math. Anal. Appl. 357(2), 349–355 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Almgren Jr. F.J.: Dirichlet’s problem for multiple valued functions and the regularity of mass minimizing integral currents. In: Minimal submanifolds and geodesics (Proc. Japan-United States Sem., Tokyo, 1977), pp. 1–6. North-Holland, Amsterdam (1979)Google Scholar
  6. 6.
    Aronszajn, N.: A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order. J. Math. Pures Appl. 36(9), 235–249 (1957)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Alessandrini, G., Vessella, S.: Local behaviour of solutions to parabolic equations. Commun. Partial Differ. Equ. 13(9), 1041–1058 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bakri, L., Casteras, J.-B.: Quantitative uniqueness for Schrödinger operator with regular potentials. Math. Methods Appl. Sci. 37(13), 1992–2008 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bakri, L.: Carleman estimates for the Schrödinger operator. Applications to quantitative uniqueness. Commun. Partial Differ. Equ. 38(1), 69–91 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bakri, L.: Quantitative uniqueness for Schrödinger operator. Indiana Univ. Math. J. 61(4), 1565–1580 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bourgain, J., Kenig, C.E.: On localization in the continuous Anderson–Bernoulli model in higher dimension. Invent. Math. 161(2), 389–426 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Camliyurt, G., Kukavica, I.: A local asymptotic expansion for a solution of the Stokes system. Evol. Equ. Control Theory 5, 647–659 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Camliyurt, G., Kukavica, I.: Quantitative unique continuation for a parabolic equation. Indiana Univ. Math. J. 67(2), 657–678 (2018)Google Scholar
  14. 14.
    Canuto, B., Rosset, E., Vessella, S.: Quantitative estimates of unique continuation for parabolic equations and inverse initial-boundary value problems with unknown boundaries. Trans. Am. Math. Soc. 354(2), 491–535 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Carleman, T.: Sur un problème d’unicité pur les systèmes d’équations aux dérivées partielles à deux variables indépendantes. Ark. Mat., Astr. Fys 26(17), 9 (1939)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Cordes, H.O.: über die eindeutige Bestimmtheit der Lösungen elliptischer Differentialgleichungen durch Anfangsvorgaben. Nachr. Akad. Wiss. Göttingen. Math.-Phys. Kl. IIa. 1956, 239–258 (1956)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Donnelly, H., Fefferman, C.: Nodal sets of eigenfunctions on Riemannian manifolds. Invent. Math. 93(1), 161–183 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Escauriaza, L.: Carleman inequalities and the heat operator. Duke Math. J. 104(1), 113–127 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Escauriaza, L., Fernández, F.J.: Unique continuation for parabolic operators. Ark. Mat. 41(1), 35–60 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Escauriaza, L., Fernández, F.J., Vessella, S.: Doubling properties of caloric functions. Appl. Anal. 85(1–3), 205–223 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Escauriaza, L., Vega, L.: Carleman inequalities and the heat operator. II. Indiana Univ. Math. J. 50(3), 1149–1169 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Escauriaza, L., Vessella, S.: Optimal three cylinder inequalities for solutions to parabolic equations with Lipschitz leading coefficients. In: Inverse problems: theory and applications (Cortona/Pisa, 2002), Contemp. Math., vol. 333, pp. 79–87. Amer. Math. Soc. Providence (2003)Google Scholar
  23. 23.
    Garofalo, N., Lin, F.-H.: Monotonicity properties of variational integrals, \(A_p\) weights and unique continuation. Indiana Univ. Math. J. 35(2), 245–268 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Hörmander, L.: Uniqueness theorems for second order elliptic differential equations. Commun. Partial Differ. Equ. 8(1), 21–64 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Ignatova, M., Kukavica, I.: Strong unique continuation for higher order elliptic equations with Gevrey coefficients. J. Differ. Equ. 252(4), 2983–3000 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Jerison, D., Kenig, C.E.: Unique continuation and absence of positive eigenvalues for Schrödinger operators. Ann. Math. (2) 121(3), 463–494 (1985). (With an appendix by E.M. Stein) MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kenig, C.E.: Restriction theorems, Carleman estimates, uniform Sobolev inequalities and unique continuation. In: Harmonic analysis and partial differential equations (El Escorial, 1987), Lecture Notes in Math., vol. 1384, pp. 69–90. Springer, Berlin (1989)Google Scholar
  28. 28.
    Kenig, C.E.: Some recent applications of unique continuation. In: Recent developments in nonlinear partial differential equations, Contemp. Math., vol. 439, pp. 25–56. Amer. Math. Soc., Providence (2007)Google Scholar
  29. 29.
    Kenig, C.E.: Quantitative unique continuation, logarithmic convexity of Gaussian means and Hardy’s uncertainty principle. In: Perspectives in partial differential equations, harmonic analysis and applications, Proc. Sympos. Pure Math., vol. 79, pp. 207–227. Amer. Math. Soc., Providence (2008)Google Scholar
  30. 30.
    Kenig, C.E., Nadirashvili, N.: A counterexample in unique continuation. Math. Res. Lett. 7(5–6), 625–630 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Kenig, C., Silvestre, L., Wang, J.-N.: On Landis’ conjecture in the plane. Commun. Partial Differ. Equ. 40(4), 766–789 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Kukavica, I.: Quantitative uniqueness for second-order elliptic operators. Duke Math. J. 91(2), 225–240 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Kukavica, I.: Quantitative uniqueness and vortex degree estimates for solutions of the Ginzburg–Landau equation. Electron. J. Differ. Equ. 2000(61), 15 (2000). (electronic) MathSciNetzbMATHGoogle Scholar
  34. 34.
    Kukavica, I.: Self-similar variables and the complex Ginzburg–Landau equation. Commun. Partial Differ. Equ. 24(3–4), 545–562 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Kurata, K.: On a backward estimate for solutions of parabolic differential equations and its application to unique continuation. In: Spectral and scattering theory and applications, Adv. Stud. Pure Math., vol. 23, pp. 247–257. Math. Soc. Japan, Tokyo (1994)Google Scholar
  36. 36.
    Koch, H., Tataru, D.: Carleman estimates and unique continuation for second-order elliptic equations with nonsmooth coefficients. Commun. Pure Appl. Math. 54(3), 339–360 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Koch, H., Tataru, D.: Carleman estimates and unique continuation for second order parabolic equations with nonsmooth coefficients. Commun. Partial Differ. Equ. 34(4–6), 305–366 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Koch, H., Tataru, D.: Sharp counterexamples in unique continuation for second order elliptic equations. J. Reine Angew. Math. 542, 133–146 (2002)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Lin, F.-H.: Nodal sets of solutions of elliptic and parabolic equations. Commun. Pure Appl. Math. 44(3), 287–308 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Mandache, N.: A counterexample to unique continuation in dimension two. Commun. Anal. Geom. 10(1), 1–10 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Meshkov, V.Z.: On the possible rate of decrease at infinity of the solutions of second-order partial differential equations. Mat. Sb. 182(3), 364–383 (1991)zbMATHGoogle Scholar
  42. 42.
    Poon, C.-C.: Unique continuation for parabolic equations. Commun. Partial Differ. Equ. 21(3–4), 521–539 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Sogge, C.D.: Strong uniqueness theorems for second order elliptic differential equations. Am. J. Math. 112(6), 943–984 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Sogge, C.D.: A unique continuation theorem for second order parabolic differential operators. Ark. Mat. 28(1), 159–182 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Vessella, S.: Carleman estimates, optimal three cylinder inequalities and unique continuation properties for parabolic operators. In: Progress in analysis (Berlin, 2001), vols. I, II, pp. 485–492. World Sci. Publ., River Edge (2003)Google Scholar
  46. 46.
    Wolff, T.H.: Note on counterexamples in strong unique continuation problems. Proc. Am. Math. Soc. 114(2), 351–356 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Wolff, T.H.: Recent work on sharp estimates in second order elliptic unique continuation problems. In: Fourier analysis and partial differential equations (Miraflores de la Sierra, 1992), Stud. Adv. Math., pp. 99–128. CRC, Boca Raton (1995)Google Scholar
  48. 48.
    Wolff, T.H.: A counterexample in a unique continuation problem. Commun. Anal. Geom. 2(1), 79–102 (1994)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Southern CaliforniaLos AngelesUSA
  2. 2.Department of MathematicsUniversity of MarylandCollege ParkUSA

Personalised recommendations