Sharp $L^p$ Carleman estimates and unique continuation

David Dos Santos Ferreira

doi:10.5802/jedp.620

Sharp

L^{p}

Carleman estimates and unique continuation

David Dos Santos Ferreira

Journées équations aux dérivées partielles (2003), article no. 6, 12 p.

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 $L^{p} - L^{p^{'}}$ boundedness properties.

MR Zbl

DOI: 10.5802/jedp.620

@incollection{JEDP_2003____A6_0,
     author = {David Dos Santos Ferreira},
     title = {Sharp $L^p$ {Carleman} estimates and unique continuation},
     booktitle = {},
     series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {6},
     pages = {1--12},
     publisher = {Universit\'e de Nantes},
     year = {2003},
     doi = {10.5802/jedp.620},
     zbl = {02079441},
     mrnumber = {2050592},
     language = {en},
     url = {https://proceedings.centre-mersenne.org/articles/10.5802/jedp.620/}
}

TY  - JOUR
AU  - David Dos Santos Ferreira
TI  - Sharp $L^p$ Carleman estimates and unique continuation
JO  - Journées équations aux dérivées partielles
PY  - 2003
SP  - 1
EP  - 12
PB  - Université de Nantes
UR  - https://proceedings.centre-mersenne.org/articles/10.5802/jedp.620/
DO  - 10.5802/jedp.620
LA  - en
ID  - JEDP_2003____A6_0
ER  -

%0 Journal Article
%A David Dos Santos Ferreira
%T Sharp $L^p$ Carleman estimates and unique continuation
%J Journées équations aux dérivées partielles
%D 2003
%P 1-12
%I Université de Nantes
%U https://proceedings.centre-mersenne.org/articles/10.5802/jedp.620/
%R 10.5802/jedp.620
%G en
%F JEDP_2003____A6_0

David Dos Santos Ferreira. Sharp $L^p$ Carleman estimates and unique continuation. Journées équations aux dérivées partielles (2003), article  no. 6, 12 p. doi : 10.5802/jedp.620. https://proceedings.centre-mersenne.org/articles/10.5802/jedp.620/

References
Cited by

[1] Brenner P., On $L^{p} - L^{p}'$ estimates for the wave equation, Math. Z., 145, 251-254, 1975. | MR | Zbl

[2] Brenner P., $L^{p} - L^{p}'$ estimates for Fourier integral operators related to hyperbolic equations, Math. Z., 152, 273-286, 1977. | MR | Zbl

[3] Dos Santos Ferreira D., Inégalités de Carleman $L^{p}$ pour des indices critiques et applications, PhD thesis, University of Rennes, 2002.

[4] Dos Santos Ferreira D., Strichartz estimates for non-selfadjoint operators and applications, to appear in Comm. PDE.

[5] Dos Santos Ferreira D., Sharp $L^{p}$ Carleman estimates and unique continuation, preprint.

[6] Escariauza L., Vega L., Carleman inequalities and the Heat operator II, Indiana Univ. Math. J., 50, 3, 2001, 1149-1169. | MR | Zbl

[7] Hörmander L., The analysis of linear partial differential operators IV, Springer-Verlag, 1985. | MR | Zbl

[8] Kapitanski L., Some generalisations of the Strichartz-Brenner inequality, Leningrad Math. J., 1, 3, 693-726, 1990. | MR | Zbl

[10] Jerison D., Kenig C.E., Unique continuation and absence of positive eigenvalues for Schrödinger operators, Adv. Math., 62, 1986, 118-134. | MR

[11] Keel M., Tao T., Endpoint Strichartz estimates, Amer. J. of Math, 120, 955-980, 1998. | MR | Zbl

[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. | MR | Zbl

[13] Koch H., Tataru D., Dispersive estimates for principally normal operators and applications to unique continuation, preprint, 2003. | MR

[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. | MR | Zbl

[15] Smith H., A parametrix construction for wave equations with $C^{1, 1}$ coefficients, J. Ann. Inst. Fourier, 48, 797-835, 1998. | Numdam | MR | Zbl

[16] Sogge C.D., Fourier integrals in classical analysis, Cambridge University Press, 1993. | MR | Zbl

[17] Sogge C.D., Oscillatory integrals, Carleman inequalities and unique continuation for second order elliptic differential equations, J. Amer. Soc., 2, 1989, 491-516. | MR | Zbl

[18] Sogge C.D., Uniqueness in Cauchy problems for hyperbolic differential operators, Trans. of AMS, 333, 2, 1992, 821-833. | MR | Zbl

[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. | Zbl

[20] Tataru D., The $X_{θ}^{s}$ spaces and unique continuation for solutions to the semilinear wave equation, Comm. PDE, 21, 1996, 841-887. | MR | Zbl

[21] Treves F., Introduction to pseudo-differential and Fourier integral operators, Plenum Press, 1980. | MR | Zbl

[22] Wolff T., Unique continuation for | Δ u| ≤ V | ∇ u| and related problems, Rev. Mat. Iberoamericana 6, 3-4, 1990, 155-200. | MR | Zbl

[23] Zuily C., Uniqueness and non-uniqueness in the Cauchy problem, Progress in Math., Birkhaüser, 1983. | Zbl

Cited by Sources: