Unique continuation for the solutions of the laplacian plus a drift

Alberto Ruiz; Luis Vega

Annales de l'institut Fourier (1991)

  • Volume: 41, Issue: 3, page 651-663
  • ISSN: 0373-0956

Abstract

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We prove unique continuation for solutions of the inequality | Δ u ( x ) | ≤ V ( x ) | ∇ u ( x ) | , x ∈ Ω a connected set contained in R n and V is in the Morrey spaces F α , p , with p ≥ ( n - 2 ) / 2 ( 1 - α ) and α < 1 . These spaces include L q for q ≥ ( 3 n - 2 ) / 2 (see [H], [BKRS]). If p = ( n - 2 ) / 2 ( 1 - α ) , the extra assumption of V being small enough is needed.

How to cite

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Ruiz, Alberto, and Vega, Luis. "Unique continuation for the solutions of the laplacian plus a drift." Annales de l'institut Fourier 41.3 (1991): 651-663. <http://eudml.org/doc/74932>.

@article{Ruiz1991,
abstract = {We prove unique continuation for solutions of the inequality $\vert \Delta u(x)\vert \le V(x)\vert \nabla u(x)\vert $, $x\in \Omega $ a connected set contained in $\{\bf R\}^ n$ and $V$ is in the Morrey spaces $F^\{\alpha ,p\}$, with $p\ge (n-2)/2(1-\alpha )$ and $\alpha &lt; 1$. These spaces include $L^ q$ for $q\ge (3n-2)/2$ (see [H], [BKRS]). If $p=(n-2)/2(1- \alpha )$, the extra assumption of $V$ being small enough is needed.},
author = {Ruiz, Alberto, Vega, Luis},
journal = {Annales de l'institut Fourier},
keywords = {Carleman -weighted inequalities; diadic decomposition; Morrey spaces},
language = {eng},
number = {3},
pages = {651-663},
publisher = {Association des Annales de l'Institut Fourier},
title = {Unique continuation for the solutions of the laplacian plus a drift},
url = {http://eudml.org/doc/74932},
volume = {41},
year = {1991},
}

TY - JOUR
AU - Ruiz, Alberto
AU - Vega, Luis
TI - Unique continuation for the solutions of the laplacian plus a drift
JO - Annales de l'institut Fourier
PY - 1991
PB - Association des Annales de l'Institut Fourier
VL - 41
IS - 3
SP - 651
EP - 663
AB - We prove unique continuation for solutions of the inequality $\vert \Delta u(x)\vert \le V(x)\vert \nabla u(x)\vert $, $x\in \Omega $ a connected set contained in ${\bf R}^ n$ and $V$ is in the Morrey spaces $F^{\alpha ,p}$, with $p\ge (n-2)/2(1-\alpha )$ and $\alpha &lt; 1$. These spaces include $L^ q$ for $q\ge (3n-2)/2$ (see [H], [BKRS]). If $p=(n-2)/2(1- \alpha )$, the extra assumption of $V$ being small enough is needed.
LA - eng
KW - Carleman -weighted inequalities; diadic decomposition; Morrey spaces
UR - http://eudml.org/doc/74932
ER -

References

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  3. [CS] S. CHANILLO, E. SAWYER, Unique continuation for ∆ + V and the C. Fefferman-Phong class, preprint. Zbl0702.35034
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  11. [K] C. KENIG, Restriction theorems, Carleman estimates, uniform Sobolev inequalities and unique continuation. Harmonic Analysis and PDE'S, Proceedings El Escorial 1987, Springer Verlag, 1384, (1989), 69-90. Zbl0685.35003
  12. [P] J. PEETRE, On the theory of Lp,λ spaces, J. Funct. Anal., 4 (1969), 71-87. Zbl0175.42602MR39 #3300
  13. [RV] A. RUIZ, L VEGA, Unique continuation for Schrödinger operators in Morrey spaces, preprint. Zbl0809.47046
  14. [St] G. STAMPACCHIA, L(p,λ)-spaces and interpolation, Comm. on Pure and Appl. Math., XVII (1964), 293-306. Zbl0149.09201MR31 #2608
  15. [Se] E. STEIN, Oscillatory integrals in Fourier Analysis. In: Beijing Lectures in Harmonic Analysis, Princeton Univ. Press, 112 (1986), 307-355. Zbl0618.42006MR88g:42022
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