$L^p$-inequalities for the laplacian and unique continuation

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

doi:10.5802/aif.843

L^{p}

-inequalities for the laplacian and unique continuation

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

Annales de l'Institut Fourier, Tome 31 (1981) no. 3, pp. 153-168.

Résumé
Abstract

Nous démontrons une inégalité de la forme

∥ | x |^{r_{f}} ∥_{L^{p} (R^{n})} \leq c (n, p, q, r) ∥ | x |^{τ + μ} Δ f ∥_{L^{q} (R^{n})} .

Comme applications nous obtenons la propriété de prolongement unique pour l’inégalité différentielle $| Δ f (x) | \leq | v (x) | | f (x) |$ si $v \in L_{L o c}^{p}$ avec $p > max (\frac{n}{2}, n - 2))$ .

We prove an inequality of the type

∥ | x |^{r_{f}} ∥_{L^{p} (R^{n})} \leq 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 $| Δ f (x) | \leq | v (x) | | f (x) |$ under suitable local integrability assumptions on the function $v$ .

MR Zbl | 4 citations dans Numdam

DOI : 10.5802/aif.843

@article{AIF_1981__31_3_153_0,
     author = {Amrein, W. O. and Berthier, A. M. and Georgescu, V.},
     title = {$L^p$-inequalities for the laplacian and unique continuation},
     journal = {Annales de l'Institut Fourier},
     pages = {153--168},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {31},
     number = {3},
     year = {1981},
     doi = {10.5802/aif.843},
     mrnumber = {83g:35011},
     zbl = {0468.35017},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.843/}
}

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Amrein, W. O.; Berthier, A. M.; Georgescu, V. $L^p$-inequalities for the laplacian and unique continuation. Annales de l'Institut Fourier, Tome 31 (1981) no. 3, pp. 153-168. doi : 10.5802/aif.843. http://www.numdam.org/articles/10.5802/aif.843/

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