Sharp L p Carleman estimates and unique continuation

David Dos Santos Ferreira

Journées équations aux dérivées partielles (2003)

  • page 1-12
  • ISSN: 0752-0360

Abstract

top
We will present a unique continuation result for solutions of second order differential equations of real principal type P ( x , D ) u + V ( x ) u = 0 with critical potential V in L n / 2 (where n is the number of variables) across non-characteristic pseudo-convex hypersurfaces. To obtain unique continuation we prove L p Carleman estimates, this is achieved by constructing a parametrix for the operator conjugated by the Carleman exponential weight and investigating its L p - L p ' boundedness properties.

How to cite

top

Dos Santos Ferreira, David. "Sharp $L^p$ Carleman estimates and unique continuation." Journées équations aux dérivées partielles (2003): 1-12. <http://eudml.org/doc/93448>.

@article{DosSantosFerreira2003,
abstract = {We will present a unique continuation result for solutions of second order differential equations of real principal type $P(x,D)u+V(x)u=0$ with critical potential $V$ in $L^\{n/2\}$ (where $n$ is the number of variables) across non-characteristic pseudo-convex hypersurfaces. To obtain unique continuation we prove $L^p$ Carleman estimates, this is achieved by constructing a parametrix for the operator conjugated by the Carleman exponential weight and investigating its $L^p-L^\{p^\{\prime \}\}$ boundedness properties.},
author = {Dos Santos Ferreira, David},
journal = {Journées équations aux dérivées partielles},
keywords = {parametrix; - boundedness},
language = {eng},
pages = {1-12},
publisher = {Université de Nantes},
title = {Sharp $L^p$ Carleman estimates and unique continuation},
url = {http://eudml.org/doc/93448},
year = {2003},
}

TY - JOUR
AU - Dos Santos Ferreira, David
TI - Sharp $L^p$ Carleman estimates and unique continuation
JO - Journées équations aux dérivées partielles
PY - 2003
PB - Université de Nantes
SP - 1
EP - 12
AB - We will present a unique continuation result for solutions of second order differential equations of real principal type $P(x,D)u+V(x)u=0$ with critical potential $V$ in $L^{n/2}$ (where $n$ is the number of variables) across non-characteristic pseudo-convex hypersurfaces. To obtain unique continuation we prove $L^p$ Carleman estimates, this is achieved by constructing a parametrix for the operator conjugated by the Carleman exponential weight and investigating its $L^p-L^{p^{\prime }}$ boundedness properties.
LA - eng
KW - parametrix; - boundedness
UR - http://eudml.org/doc/93448
ER -

References

top
  1. [1] Brenner P., On L p - L p ' estimates for the wave equation, Math. Z., 145, 251-254, 1975. Zbl0321.35052MR387819
  2. [2] Brenner P., L p - L p ' estimates for Fourier integral operators related to hyperbolic equations, Math. Z., 152, 273-286, 1977. Zbl0325.35009MR430872
  3. [3] Dos Santos Ferreira D., Inégalités de Carleman L p pour des indices critiques et applications, PhD thesis, University of Rennes, 2002. 
  4. [4] Dos Santos Ferreira D., Strichartz estimates for non-selfadjoint operators and applications, to appear in Comm. PDE. 
  5. [5] Dos Santos Ferreira D., Sharp L p Carleman estimates and unique continuation, preprint. 
  6. [6] Escariauza L., Vega L., Carleman inequalities and the Heat operator II, Indiana Univ. Math. J., 50, 3, 2001, 1149-1169. Zbl1029.35046MR1871351
  7. [7] Hörmander L., The analysis of linear partial differential operators IV, Springer-Verlag, 1985. Zbl0612.35001MR781537
  8. [8] Kapitanski L., Some generalisations of the Strichartz-Brenner inequality, Leningrad Math. J., 1, 3, 693-726, 1990. Zbl0732.35118MR1015129
  9. [10] Jerison D., Kenig C.E., Unique continuation and absence of positive eigenvalues for Schrödinger operators, Adv. Math., 62, 1986, 118-134. MR865834
  10. [11] Keel M., Tao T., Endpoint Strichartz estimates, Amer. J. of Math, 120, 955-980, 1998. Zbl0922.35028MR1646048
  11. [12] Koch H., Tataru D., Carleman estimates and unique continuation for second order elliptic equations with non-smooth coefficients, Comm. Pure Appl. Math., 54, 3, 339-360, 2001. Zbl1033.35025MR1809741
  12. [13] Koch H., Tataru D., Dispersive estimates for principally normal operators and applications to unique continuation, preprint, 2003. Zbl1055.35152MR2056851
  13. [14] Kenig C.E., Ruiz A., Sogge C.D., Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators, Duke Math. J., 55, 2, 1987, 329-347. Zbl0644.35012MR894584
  14. [15] Smith H., A parametrix construction for wave equations with C 1 , 1 coefficients, J. Ann. Inst. Fourier, 48, 797-835, 1998. Zbl0974.35068MR1644105
  15. [16] Sogge C.D., Fourier integrals in classical analysis, Cambridge University Press, 1993. Zbl0783.35001MR1205579
  16. [17] Sogge C.D., Oscillatory integrals, Carleman inequalities and unique continuation for second order elliptic differential equations, J. Amer. Soc., 2, 1989, 491-516. Zbl0703.35027MR999662
  17. [18] Sogge C.D., Uniqueness in Cauchy problems for hyperbolic differential operators, Trans. of AMS, 333, 2, 1992, 821-833. Zbl0763.35012MR1066449
  18. [19] Strichartz R.S., Restriction of Fourier transform to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44, 1977, 705-774. Zbl0372.35001
  19. [20] Tataru D., The X θ s spaces and unique continuation for solutions to the semilinear wave equation, Comm. PDE, 21, 1996, 841-887. Zbl0853.35017MR1391526
  20. [21] Treves F., Introduction to pseudo-differential and Fourier integral operators, Plenum Press, 1980. Zbl0453.47027MR597145
  21. [22] Wolff T., Unique continuation for | Δ u| ≤ V | ∇ u| and related problems, Rev. Mat. Iberoamericana 6, 3-4, 1990, 155-200. Zbl0735.35024MR1125760
  22. [23] Zuily C., Uniqueness and non-uniqueness in the Cauchy problem, Progress in Math., Birkhaüser, 1983. Zbl0521.35003

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.