Carleman estimates for a subelliptic operator and unique continuation

Garofalo, Nicola ; Shen, Zhongwei

Annales de l'Institut Fourier, Tome 44 (1994), p. 129-166 / Harvested from Numdam

Résumé

Nous démontrons une inéqualité du type de Carleman pour l’opérateur sous-elliptique de la forme ${\cal L}=\Delta _z + \vert z\vert ^2 \partial ^2_t$ dans ${\Bbb R}^{n+1}$ avec $n\ge 2$, $z\in {\Bbb R}^n$, et $t\in {\Bbb R}$. On en déduit que $-{\cal L}+V$ possède la propriété d’unicité stricte du prolongement des solutions aux points $(0, t)$, $t\in {\Bbb R}$, si le potentiel $V$ appartient localement à des espaces $L^p$ particuliers.

Abstract

We establish a Carleman type inequality for the subelliptic operator ${\cal L}=\Delta _z + \vert x\vert ^2 \partial ^2_t$ in ${\Bbb R}^{n+1}$, $n\ge 2$, where $z\in {\Bbb R}^n$, $t\in {\Bbb R}$. As a consequence, we show that $-{\cal L}+V$ has the strong unique continuation property at points of the degeneracy manifold $\lbrace (0, t)\in {\Bbb R}^{n+1}\vert t\in {\Bbb R}\rbrace $ if the potential $V$ is locally in certain $L^p$ spaces.

Détail

Publié le : 1994-01-01

MR 94m:35037 | Zbl 0791.35017

DOI : https://doi.org/10.5802/aif.1392

Bibliographie

[ABV]W. O. Amrein, A. M. Berthier and V. Georgescu, Lp inequalities for the Laplacian and unique continuation, Ann. Inst. Fourier, Grenoble, 31-3 (1981), 153-168. | Numdam | MR 83g:35011 | Zbl 0468.35017

[B]H. Bahouri, Non-prolongement unique des solutions d'opérateurs, “Somme de carrés”, Ann. Inst. Fourier, Grenoble, 36-4 (1986), 137-155. | Numdam | MR 88c:35027 | Zbl 0603.35008

[C]T. Carleman, Sur un problème d'unicité pour les systèmes d'èquations aux derivées partielles à deux variables indépendantes, Ark. Mat., 26B (1939), 1-9. | MR 1,55f | Zbl 0022.34201

[E]A. Erdelyi (Director), Higher transcendental functions, Bateman manuscript project, McGraw-Hill, New York, 1955. | MR 16,586c | Zbl 0064.06302

[G]N. Garofalo, Unique continuation for a class of elliptic operators which degenerate on a manifold of arbitrary codimension, J. Diff. Eq., 104 (1) (1993), 117-146. | MR 94i:35037 | Zbl 0788.35051

[Gr]P. Greiner, Spherical harmonics on the Heisenberg group, Canad. Math. Bull., 23(4) (1980), 383-396. | MR 82e:43009 | Zbl 0496.22012

[Gru1]V.V. Grushin, On a class of hypoelliptic operators, Math. USSR Sbornik, 12(3) (1970), 458-476. | Zbl 0252.35057

[Gru2]V.V. Grushin, On a class of hypoelliptic pseudodifferential operators degenerate on a submanifold, Math. USSR Sbornik, 13(2) (1971), 155-186. | Zbl 0238.47038

[H]L. Hörmander, Uniqueness theorems for second order elliptic differential equations, Comm. P. D. E., 8 (1983), 21-64. | MR 85c:35018 | Zbl 0546.35023

[J]D. Jerison, Carleman inequalities for the Dirac and Laplace operators and unique continuation, Adv. in Math., 63 (1986), 118-134. | MR 88b:35218 | Zbl 0627.35008

[JK]D. Jerison and C. E. Kenig, Unique continuation and absence of positive eigenvalues for Schrödinger operators, Ann. of Math., 121 (1985), 463-494. | MR 87a:35058 | Zbl 0593.35119

[K]C.E. Kenig, Restriction theorems, Carleman estimates, uniform Sobolev inequalities and unique continuation, Lecture Notes in Math., 1384 (1989) 69-90. | Zbl 0685.35003

[RS]L.P. Rothschild and E.M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math., 137 (1976), 247-320. | MR 55 #9171 | Zbl 0346.35030

[SS]M. Schechter and B. Simon, Unique continuation for Schrödinger operators with unbounded potential, J. Math. Anal. Appl., 77 (1980), 482-492. | MR 83b:35031 | Zbl 0458.35024

[S]C.D. Sogge, Oscillatory integrals and spherical harmonics, Duke Math. J., 53 (1986), 43-65. | MR 87g:42026 | Zbl 0636.42018

[SW]E.M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, 1971. | MR 46 #4102 | Zbl 0232.42007

[Sz] G. Szegö, Orthogonal Polynomials, A. M. S. Colloq. Publ., 4th edition, 23, 1975.