A note of uniqueness on the Cauchy problem for Schrödinger or heat equations with degenerate elliptic principal parts

Hideki Takuwa

Bollettino dell'Unione Matematica Italiana (2004)

  • Volume: 7-B, Issue: 3, page 731-744
  • ISSN: 0392-4041

Abstract

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We study the local uniqueness in the Cauchy problem for Schrödinger or heat equations whose principal parts are nonnegative. We show the compact uniqueness under a weak form of pseudo convexity. This makes up for the known results under the conormal pseudo convexity given by Tataru, Hörmander, Robbiano- Zuily and L. T'Joen. Our method is based on a kind of integral transform and a weak form of Carleman estimate for degenerate elliptic operators.

How to cite

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Takuwa, Hideki. "A note of uniqueness on the Cauchy problem for Schrödinger or heat equations with degenerate elliptic principal parts." Bollettino dell'Unione Matematica Italiana 7-B.3 (2004): 731-744. <http://eudml.org/doc/195272>.

@article{Takuwa2004,
abstract = {We study the local uniqueness in the Cauchy problem for Schrödinger or heat equations whose principal parts are nonnegative. We show the compact uniqueness under a weak form of pseudo convexity. This makes up for the known results under the conormal pseudo convexity given by Tataru, Hörmander, Robbiano- Zuily and L. T'Joen. Our method is based on a kind of integral transform and a weak form of Carleman estimate for degenerate elliptic operators.},
author = {Takuwa, Hideki},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {731-744},
publisher = {Unione Matematica Italiana},
title = {A note of uniqueness on the Cauchy problem for Schrödinger or heat equations with degenerate elliptic principal parts},
url = {http://eudml.org/doc/195272},
volume = {7-B},
year = {2004},
}

TY - JOUR
AU - Takuwa, Hideki
TI - A note of uniqueness on the Cauchy problem for Schrödinger or heat equations with degenerate elliptic principal parts
JO - Bollettino dell'Unione Matematica Italiana
DA - 2004/10//
PB - Unione Matematica Italiana
VL - 7-B
IS - 3
SP - 731
EP - 744
AB - We study the local uniqueness in the Cauchy problem for Schrödinger or heat equations whose principal parts are nonnegative. We show the compact uniqueness under a weak form of pseudo convexity. This makes up for the known results under the conormal pseudo convexity given by Tataru, Hörmander, Robbiano- Zuily and L. T'Joen. Our method is based on a kind of integral transform and a weak form of Carleman estimate for degenerate elliptic operators.
LA - eng
UR - http://eudml.org/doc/195272
ER -

References

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  1. COLOMBINI, F.- DEL SANTO, D.- ZUILY, C., Uniqueness and non-uniqueness in the Cauchy problem for a degenerate elliptic operator, Amer. J. Math., 115 (1993), 1281-1297. Zbl0792.35065MR1254734
  2. HÖRMANDER, L., Linear Partial Differential Operators, Springer Verlag, 1963. Zbl0108.09301MR404822
  3. HÖRMANDER, L., The Analysis of Linear Partial Differential Operators, Springer Verlag, 1983-1985. Zbl0521.35002
  4. HÖRMANDER, L., On the uniqueness of the Cauchy problem under partial analyticity assumptions, Geometrical Optics and Related Topics, Birkhäuser, 1997, 179-219. Zbl0907.35002MR2033496
  5. T'JOEN, L., Uniqueness in the Cauchy problem for quasi-homogeneous operators with partially holomorphic coefficients, Osaka J. Math., 37 (2000), 925-951. Zbl0982.35005MR1809913
  6. NIRENBERG, L., Uniqueness in the Cauchy problem for a degenerate elliptic second order equation, Differential Geometry and Complex Analysis, Springer-Verlag, 1985, 213-218 Zbl0572.35043MR780047
  7. ROBBIANO, L., Théorème d'unicité adapté au contrôle des solutions des problèmes hyperboliques, Comm. Partial Differential Equations, 16 (1991), 789-800. Zbl0735.35086MR1113107
  8. ROBBIANO, L.- ZUILY, C., Uniqueness in the Cauchy problem for operators with partially holomorphic coefficients, Invent. Math., 131 (1998), 493-539. Zbl0909.35004MR1614547
  9. DEL SANTO, D., A Carleman estimate for degenerate elliptic operators with an application to an illposed problem, Boll. U. M. I., 11-B (1997), 327-339. Zbl0877.35051MR1459281
  10. TARAMA, S., On the estimate of some conjugation, preprint. 
  11. TATARU, D., Unique continuation for solutions to PDE's; between Hörmander's theorem and Holmgren's theorem, Comm. Partial Differential Equations, 20 (1995), 855-884. Zbl0846.35021
  12. ZUILY, C., Uniqueness and Non-uniqueness in the Cauchy Problem, Progress in Math.33, Birkhäuser, 1983. Zbl0521.35003

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