Unique continuation for the solutions of the laplacian plus a drift
Annales de l'Institut Fourier, Tome 41 (1991) no. 3, pp. 651-663.

Nous prouvons l’unicité du prolongement pour les solutions de l’inégalité |Δu(x)|V(x)|u(x)|, xΩΩ est une partie connexe de R n et V appartient aux espaces de Morrey F α,p , avec p(n-2)/2(1-α) et α<1. Ces espaces contiennent L q pour q(3n-2)/2 (voir L. Hörmander, Comm. PDE, 8 (1983, 21-64 et Barceló, Kenig, Ruiz, Sogge, Ill. J. of Math., 32-2 (1988), 230-245).

We prove unique continuation for solutions of the inequality |Δu(x)|V(x)|u(x)|, xΩ 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(3n-2)/2 (see [H], [BKRS]). If p=(n-2)/2(1-α), the extra assumption of V being small enough is needed.

@article{AIF_1991__41_3_651_0,
     author = {Ruiz, Alberto and Vega, Luis},
     title = {Unique continuation for the solutions of the laplacian plus a drift},
     journal = {Annales de l'Institut Fourier},
     pages = {651--663},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {41},
     number = {3},
     year = {1991},
     doi = {10.5802/aif.1268},
     zbl = {0772.35008},
     mrnumber = {92k:35043},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1268/}
}
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
SP  - 651
EP  - 663
VL  - 41
IS  - 3
PB  - Institut Fourier
PP  - Grenoble
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1268/
DO  - 10.5802/aif.1268
LA  - en
ID  - AIF_1991__41_3_651_0
ER  - 
%0 Journal Article
%A Ruiz, Alberto
%A Vega, Luis
%T Unique continuation for the solutions of the laplacian plus a drift
%J Annales de l'Institut Fourier
%D 1991
%P 651-663
%V 41
%N 3
%I Institut Fourier
%C Grenoble
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1268/
%R 10.5802/aif.1268
%G en
%F AIF_1991__41_3_651_0
Ruiz, Alberto; Vega, Luis. Unique continuation for the solutions of the laplacian plus a drift. Annales de l'Institut Fourier, Tome 41 (1991) no. 3, pp. 651-663. doi : 10.5802/aif.1268. https://aif.centre-mersenne.org/articles/10.5802/aif.1268/

[BKRS] Barceló, Kenig, Ruiz, Sogge, Weighted Sobolev inequalities and unique continuation for the Laplaciaan plus lower order terms, III. J. of Math., 32, n.2 (1988), 230-245. | MR | Zbl

[C] S. Campanato, Proprietá di inclusione per spazi di Morrey, Ricerche Mat., 12 (1963), 67-896. | MR | Zbl

[CS] S. Chanillo, E. Sawyer, Unique continuation for ∆ + V and the C. Fefferman-Phong class, preprint. | Zbl

[ChR] F. Chiarenza, A. Ruiz, Uniform L2 weighted inequalities, Proc. A.M.S., to appear. | Zbl

[ChF] F. Chiarenza, M. Frasca, A remark on a paper by C. Fefferman, Proc. A.M.S., (Feb. 1990), 407-409. | MR | Zbl

[FeP] C. Fefferman, D.H. Phong, Lower bounds for Schrödinger equations, Journées Eqs. aux deriv. partielles, St. Jean de Monts, 1982. | Numdam | Zbl

[GL] N. Garofalo, F.H. Lin, Unique continuation for elliptic operators; a geometric variational approach, Comm. Pure Appl. Math., 40 (1987), 347-366. | MR | Zbl

[H] L. Hörmander, Uniqueness theorem for second order differential operators, Comm. PDE, 8 (1983), 21-64. | Zbl

[Je] D. Jerison, Carleman inequalities for the Dirac and Laplace operators and unique continuation, Adv. In Math., 63 (1986), 118-134. | MR | Zbl

[Jo] J.L. Journé, Calderón-Zygmund operators, pseudo-differential operators, and the Cauchyintegral of Calderón, Lecture Notes in Math., Springer Verlag, 1983. | Zbl

[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. | Zbl

[P] J. Peetre, On the theory of Lp,λ spaces, J. Funct. Anal., 4 (1969), 71-87. | MR | Zbl

[RV] A. Ruiz, L Vega, Unique continuation for Schrödinger operators in Morrey spaces, preprint. | Zbl

[St] G. Stampacchia, L(p,λ)-spaces and interpolation, Comm. on Pure and Appl. Math., XVII (1964), 293-306. | MR | Zbl

[Se] E. Stein, Oscillatory integrals in Fourier Analysis. In: Beijing Lectures in Harmonic Analysis, Princeton Univ. Press, 112 (1986), 307-355. | MR | Zbl

[T] P. Tomas, A restriction theorem for the Fourier transform, Bull. AMS, (1975), 477-478. | MR | Zbl

Cité par Sources :