Three cylinder inequalities and unique continuation properties for parabolic equations

Sergio Vessella

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2002)

  • Volume: 13, Issue: 2, page 107-120
  • ISSN: 1120-6330

Abstract

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We prove the following unique continuation property. Let u be a solution of a second order linear parabolic equation and S a segment parallel to the t -axis. If u has a zero of order faster than any non constant and time independent polynomial at each point of S then u vanishes in each point, x , t ′ , such that the plane t = t ′ has a non empty intersection with S .

How to cite

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Vessella, Sergio. "Three cylinder inequalities and unique continuation properties for parabolic equations." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 13.2 (2002): 107-120. <http://eudml.org/doc/252330>.

@article{Vessella2002,
abstract = {We prove the following unique continuation property. Let $u$ be a solution of a second order linear parabolic equation and $S$ a segment parallel to the $t$-axis. If $u$ has a zero of order faster than any non constant and time independent polynomial at each point of $S$ then $u$ vanishes in each point, $(x,t^\{\prime\})$, such that the plane $t = t^\{\prime\}$ has a non empty intersection with $S$.},
author = {Vessella, Sergio},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Continuation of solutions; Stability estimates; Ill-posed Problem; continuation of solutions; stability estimates; ill-posed problem},
language = {eng},
month = {6},
number = {2},
pages = {107-120},
publisher = {Accademia Nazionale dei Lincei},
title = {Three cylinder inequalities and unique continuation properties for parabolic equations},
url = {http://eudml.org/doc/252330},
volume = {13},
year = {2002},
}

TY - JOUR
AU - Vessella, Sergio
TI - Three cylinder inequalities and unique continuation properties for parabolic equations
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2002/6//
PB - Accademia Nazionale dei Lincei
VL - 13
IS - 2
SP - 107
EP - 120
AB - We prove the following unique continuation property. Let $u$ be a solution of a second order linear parabolic equation and $S$ a segment parallel to the $t$-axis. If $u$ has a zero of order faster than any non constant and time independent polynomial at each point of $S$ then $u$ vanishes in each point, $(x,t^{\prime})$, such that the plane $t = t^{\prime}$ has a non empty intersection with $S$.
LA - eng
KW - Continuation of solutions; Stability estimates; Ill-posed Problem; continuation of solutions; stability estimates; ill-posed problem
UR - http://eudml.org/doc/252330
ER -

References

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  2. Aronszajn, N. - Krzywicki, A. - Szarski, J., A unique continuation theorem for exterior differential forms on Riemannian manifold. Ark. for Matematik, 4, (34), 1962, 417-453. Zbl0107.07803MR140031
  3. Canuto, B. - Rosset, E. - Vessella, S., Quantitative estimates of unique continuation for parabolic equations and inverse-initial boundary value problems with unknown boundaries. Transactions of AMS, to appear. Zbl0992.35112MR1862557DOI10.1090/S0002-9947-01-02860-4
  4. Canuto, B. - Rosset, E. - Vessella, S., A stability result in the localization of cavities in a thermic conducting medium. Preprint n. 59, 2001, Laboratoire de Mathématiques Appliquées, Université de Versailles. Zbl1225.35255MR1925040DOI10.1051/cocv:2002066
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  6. Glagoleva, R.Ja., Some properties of solutions of a linear second order parabolic equation. Math. USSR-Sbornik, 3 (1), 1967, 41-67. Zbl0172.14601
  7. Hörmander, L., Uniqueness theorem for second order elliptic differential equations. Comm. Part. Diff. Equations, 8 (1), 1983, 21-64. Zbl0815.35063
  8. Lees, M. - Protter, M.H., Unique continuation for parabolic equations. Duke Math. J., 28, 1961, 369-382. Zbl0143.33301MR140840
  9. Lin, F.H., A uniqueness theorem for parabolic equations. Comm. Pure Appl. Math., XLIII, 1990, 127-136. Zbl0727.35063MR1024191DOI10.1002/cpa.3160430105
  10. Varin, A.A., Three-Cylinder theorem for certain class of semilinear parabolic equations. Mat. Zametki, 51, (1), 1992, 32-41. Zbl0780.35050MR1165278DOI10.1007/BF01229430
  11. Vessella, S., Carleman estimates, optimal three cylinder inequality and unique continuation properties for solutions to parabolic equations. Quaderno DiMaD, novembre 2001, 1-12. Zbl1024.35021

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