Carleman estimates for a subelliptic operator and unique continuation

Nicola Garofalo; Zhongwei Shen

Annales de l'institut Fourier (1994)

  • Volume: 44, Issue: 1, page 129-166
  • ISSN: 0373-0956

Abstract

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We establish a Carleman type inequality for the subelliptic operator ℒ = Δ z + | x | 2 ∂ t 2 in ℝ n + 1 , n ≥ 2 , where z ∈ ℝ n , t ∈ ℝ . As a consequence, we show that - ℒ + V has the strong unique continuation property at points of the degeneracy manifold { ( 0 , t ) ∈ ℝ n + 1 | t ∈ ℝ } if the potential V is locally in certain L p spaces.

How to cite

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Garofalo, Nicola, and Shen, Zhongwei. "Carleman estimates for a subelliptic operator and unique continuation." Annales de l'institut Fourier 44.1 (1994): 129-166. <http://eudml.org/doc/75052>.

@article{Garofalo1994,
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.},
author = {Garofalo, Nicola, Shen, Zhongwei},
journal = {Annales de l'institut Fourier},
keywords = {Grushin operator; Carleman type inequality; subelliptic operator; strong unique continuation property},
language = {eng},
number = {1},
pages = {129-166},
publisher = {Association des Annales de l'Institut Fourier},
title = {Carleman estimates for a subelliptic operator and unique continuation},
url = {http://eudml.org/doc/75052},
volume = {44},
year = {1994},
}

TY - JOUR
AU - Garofalo, Nicola
AU - Shen, Zhongwei
TI - Carleman estimates for a subelliptic operator and unique continuation
JO - Annales de l'institut Fourier
PY - 1994
PB - Association des Annales de l'Institut Fourier
VL - 44
IS - 1
SP - 129
EP - 166
AB - 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.
LA - eng
KW - Grushin operator; Carleman type inequality; subelliptic operator; strong unique continuation property
UR - http://eudml.org/doc/75052
ER -

References

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  1. [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. Zbl0468.35017MR83g:35011
  2. [B]H. BAHOURI, Non-prolongement unique des solutions d'opérateurs, “Somme de carrés”, Ann. Inst. Fourier, Grenoble, 36-4 (1986), 137-155. Zbl0603.35008MR88c:35027
  3. [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. Zbl0022.34201MR1,55fJFM65.0394.03
  4. [E]A. ERDELYI (Director), Higher transcendental functions, Bateman manuscript project, McGraw-Hill, New York, 1955. Zbl0064.06302MR16,586c
  5. [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. Zbl0788.35051MR94i:35037
  6. [Gr]P. GREINER, Spherical harmonics on the Heisenberg group, Canad. Math. Bull., 23(4) (1980), 383-396. Zbl0496.22012MR82e:43009
  7. [Gru1]V.V. GRUSHIN, On a class of hypoelliptic operators, Math. USSR Sbornik, 12(3) (1970), 458-476. Zbl0252.35057
  8. [Gru2]V.V. GRUSHIN, On a class of hypoelliptic pseudodifferential operators degenerate on a submanifold, Math. USSR Sbornik, 13(2) (1971), 155-186. Zbl0238.47038
  9. [H]L. HÖRMANDER, Uniqueness theorems for second order elliptic differential equations, Comm. P. D. E., 8 (1983), 21-64. Zbl0546.35023MR85c:35018
  10. [J]D. JERISON, Carleman inequalities for the Dirac and Laplace operators and unique continuation, Adv. in Math., 63 (1986), 118-134. Zbl0627.35008MR88b:35218
  11. [JK]D. JERISON and C. E. KENIG, Unique continuation and absence of positive eigenvalues for Schrödinger operators, Ann. of Math., 121 (1985), 463-494. Zbl0593.35119MR87a:35058
  12. [K]C.E. KENIG, Restriction theorems, Carleman estimates, uniform Sobolev inequalities and unique continuation, Lecture Notes in Math., 1384 (1989) 69-90. Zbl0685.35003
  13. [RS]L.P. ROTHSCHILD and E.M. STEIN, Hypoelliptic differential operators and nilpotent groups, Acta Math., 137 (1976), 247-320. Zbl0346.35030MR55 #9171
  14. [SS]M. SCHECHTER and B. SIMON, Unique continuation for Schrödinger operators with unbounded potential, J. Math. Anal. Appl., 77 (1980), 482-492. Zbl0458.35024MR83b:35031
  15. [S]C.D. SOGGE, Oscillatory integrals and spherical harmonics, Duke Math. J., 53 (1986), 43-65. Zbl0636.42018MR87g:42026
  16. [SW]E.M. STEIN and G. WEISS, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, 1971. Zbl0232.42007MR46 #4102
  17. [Sz] G. SZEGÖ, Orthogonal Polynomials, A. M. S. Colloq. Publ., 4th edition, 23, 1975. 

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