L p -inequalities for the laplacian and unique continuation

W. O. Amrein; A. M. Berthier; V. Georgescu

Annales de l'institut Fourier (1981)

  • Volume: 31, Issue: 3, page 153-168
  • ISSN: 0373-0956

Abstract

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We prove an inequality of the type | x | r f L p ( R n ) c ( n , p , q , r ) | x | τ + μ Δ f L q ( R n ) . This is then used to derive the unique continuation property for the differential inequality | Δ f ( x ) | | v ( x ) | | f ( x ) | under suitable local integrability assumptions on the function v .

How to cite

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Amrein, W. O., Berthier, A. M., and Georgescu, V.. "$L^p$-inequalities for the laplacian and unique continuation." Annales de l'institut Fourier 31.3 (1981): 153-168. <http://eudml.org/doc/74502>.

@article{Amrein1981,
abstract = {We prove an inequality of the type\begin\{\}\Vert |x|^\{r\_f\}\Vert \_\{L^p(\{\bf R\}^n)\} \le c(n,p,q,r)\Vert |x|^\{\tau +\mu \} \Delta f\Vert \_\{L^q(\{\bf R\}^n)\}.\end\{\}This is then used to derive the unique continuation property for the differential inequality $|\Delta f(x)| \le |v(x)|\ |f(x)|$ under suitable local integrability assumptions on the function $v$.},
author = {Amrein, W. O., Berthier, A. M., Georgescu, V.},
journal = {Annales de l'institut Fourier},
keywords = {unique continuation properties; differential inequality; non-existence of positive eigenvalues; self-adjoint Schrödinger operators},
language = {eng},
number = {3},
pages = {153-168},
publisher = {Association des Annales de l'Institut Fourier},
title = {$L^p$-inequalities for the laplacian and unique continuation},
url = {http://eudml.org/doc/74502},
volume = {31},
year = {1981},
}

TY - JOUR
AU - Amrein, W. O.
AU - Berthier, A. M.
AU - Georgescu, V.
TI - $L^p$-inequalities for the laplacian and unique continuation
JO - Annales de l'institut Fourier
PY - 1981
PB - Association des Annales de l'Institut Fourier
VL - 31
IS - 3
SP - 153
EP - 168
AB - We prove an inequality of the type\begin{}\Vert |x|^{r_f}\Vert _{L^p({\bf R}^n)} \le c(n,p,q,r)\Vert |x|^{\tau +\mu } \Delta f\Vert _{L^q({\bf R}^n)}.\end{}This is then used to derive the unique continuation property for the differential inequality $|\Delta f(x)| \le |v(x)|\ |f(x)|$ under suitable local integrability assumptions on the function $v$.
LA - eng
KW - unique continuation properties; differential inequality; non-existence of positive eigenvalues; self-adjoint Schrödinger operators
UR - http://eudml.org/doc/74502
ER -

References

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  1. [1] R.A. ADAMS, Sobolev Spaces, Academic Press, New York, 1975. Zbl0314.46030MR56 #9247
  2. [2] A.M. BERTHIER, Sur le spectre ponctuel de l'opérateur de Schrödinger, C.R. Acad. Sci., Paris 290 A, (1980), 393-395 ; On the Point Spectrum of Schrödinger Operators, Ann. Sci. Ecole Normale Supérieure (to appear). Zbl0454.35070MR81a:35072
  3. [3] N. DUNFORD and J.T. SCHWARTZ, Linear Operators, Part I, Interscience, New York, 1957. 
  4. [4] V. GEORGESCU, On the Unique Continuation Property for Schrödinger Hamiltonians, Helv. Phys. Acta, 52 (1979), 655-670. 
  5. [5] G.H. HARDY, J.E. LITTELEWOOD and G. POLYA, Inequalities, Cambridge University Press, 1952. Zbl0047.05302
  6. [6] E. HEINZ, Über die Eindeutigkeit beim Cauchy'schen Anfangswert-problem einer elliptischen Differentialgleichung zweiter Ordnung, Nachr. Akad.-Wiss. Göttingen, II (1955), 1-12. Zbl0067.07503MR17,626c
  7. [7] L. HÖRMANDER, Linear Partial Differential Operators, Springer, Berlin, 1963. Zbl0108.09301
  8. [8] M. SCHECHTER and B. SIMON, Unique Continuation for Schrödinger Operators with Unbounded Potentials, J. Math. Anal. Appl., 77 (1980), 482-492. Zbl0458.35024MR83b:35031
  9. [9] E.M. STEIN and G. WEISS, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, 1971. Zbl0232.42007MR46 #4102
  10. [10] H. TRIEBEL, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978. Zbl0387.46032

Citations in EuDML Documents

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  1. Eric T. Sawyer, Unique continuation for Schrödinger operators in dimension three or less
  2. M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, Jörg Swetina, Pointwise bounds on the asymptotics of spherically averaged L 2 -solutions of one-body Schrödinger equations
  3. Werner O. Amrein, Bornes inférieurs pour des fonctions propres de l'opérateur de Schrödinger
  4. Nicola Garofalo, Zhongwei Shen, Carleman estimates for a subelliptic operator and unique continuation
  5. Lech Zieliński, Charge transfer scatteringin a constant electric field

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