Sharp $L^p$ Carleman estimates and unique continuation

Dos Santos Ferreira, David

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

We will present a unique continuation result for solutions of second order differential equations of real principal type $P(x,D)u+V\left(x\right)u=0$ 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 $L^p-L^{p^{\text{'}}}$ boundedness properties.

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

[2] Brenner P., $L^p-L^p\text{'}$ 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 $X_{\theta }^s$ 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