English
$L^p$-theory for Schrödinger systems
Multi angle
Auteurs : Rhandi, Abdelaziz (Auteur de la Conférence)
CIRM (Editeur )
35J15 - General theory of second-order, elliptic equations
47D06 - One-parameter semigroups and linear evolution equations
47D08 - Schrödinger and Feynman-Kac semigroups
35J47 - Second-order elliptic systems
CIRM (Editeur )
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Résumé :
In this talk we study for $p\in \left ( 1,\infty \right )$ the $L^{p}$-realization of the vector-valued Schrödinger operator $\mathcal{L}u:= div\left ( Q\triangledown u \right )+Vu$. Using a noncommutative version of the Dore–Venni theorem due to Monniaux and Prüss, and a perturbation theorem by Okazawa, we prove that $L^{p}$, the $L^{p}$-realization of $\mathcal{L}$, defined on the intersection of the natural domains of the differential and multiplication operators which form $\mathcal{L}$, generates a strongly continuous contraction semigroup on $L^{p}\left ( \mathbb{R}^{d} ;\mathbb{C}^{m}\right )$. We also study additional properties of the semigroup such as positivity, ultracontractivity, Gaussian estimates and compactness of the resolvent. We end the talk by giving some generalizations obtained recently and several examples.
Keywords : system of PDE; Schrödinger operator; strongly continuous semigroup
Codes MSC : Keywords : system of PDE; Schrödinger operator; strongly continuous semigroup
35J15 - General theory of second-order, elliptic equations
47D06 - One-parameter semigroups and linear evolution equations
47D08 - Schrödinger and Feynman-Kac semigroups
35J47 - Second-order elliptic systems
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Informations sur la Vidéo
Réalisateur : Hennenfent, GuillaumeLangue : Anglais Date de publication : 29/11/2019 Date de captation : 28/10/2019 Sous collection : Research talks arXiv category : Analysis of PDEs Domaine : PDE ; Dynamical Systems & ODE Format : MP4 (.mp4) - HD Durée : 00:28:42 Audience : Researchers Download : https://videos.cirm-math.fr/2019-10-31_Rhandi.mp4 
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Informations sur la Rencontre
Nom de la rencontre : Evolution Equations: Applied and Abstract Perspectives / Equations d'évolution: perspectives appliquées et abstraitesOrganisateurs de la rencontre : Disser, Karoline ; Haller-Dintelmann, Robert ; Kyed, Mads ; Saal, Jürgen Dates : 28/10/2019 - 01/11/2019 Année de la rencontre : 2019 URL Congrès : https://conferences.cirm-math.fr/2071.html
Données de citation
DOI : 10.24350/CIRM.V.19576003Citer cette vidéo: Rhandi, Abdelaziz (2019). $L^p$-theory for Schrödinger systems . CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19576003 URI : http://dx.doi.org/10.24350/CIRM.V.19576003 |
Voir aussi
- [Multi angle] Determining the global dynamics of the two-dimensional Navier-Stokes equations by a scalar ODE / Auteur de la Conférence Titi, Edriss S..
- [Multi angle] Global well-posedness for the primitive equations coupled to nonlinear moisture dynamics with phase changes / Auteur de la Conférence Li, Jinkai.
- [ Post-edited] $L^{r}$-Helmholtz-Weyl decomposition in 3D exterior domains / Auteur de la Conférence Kozono, Hideo.
- [Multi angle] Dispersive decay for small localized solutions to the Korteweg-de Vries equation / Auteur de la Conférence Koch, Herbert.
Bibliographie
- KUNZE, Markus, LORENZI, Luca, MAICHINE, Abdallah, et al. ${L^ p} $-theory for Schr\" odinger systems. arXiv preprint arXiv:1705.03333, 2017. - https://arxiv.org/abs/1705.03333
- HIEBER, Matthias, LORENZI, Luca, PRÜSS, Jan, et al. Global properties of generalized Ornstein–Uhlenbeck operators on Lp (RN, RN) with more than linearly growing coefficients. Journal of Mathematical Analysis and Applications, 2009, vol. 350, no 1, p. 100-121. - http://dx.doi.org/10.1016/j.jmaa.2008.09.011
- KUNZE, M., LORENZI, L., MAICHINE, A., et al. Lp-theory for Schrödinger systems, Math. Nachr, vol. 292 n°8 p1763-1776 - https://doi.org/10.1002/mana.201800206
- KUNZE, Markus, MAICHINE, Abdallah, et RHANDI, Abdelaziz. Vector-valued Schr\" odinger operators on $ L^ p $-spaces. arXiv preprint arXiv:1802.09771, 2018. - https://arxiv.org/pdf/1802.09771.pdf
- MAICHINE, Abdallah et RHANDI, Abdelaziz. On a polynomial scalar perturbation of a Schrödinger system in Lp-spaces. Journal of Mathematical Analysis and Applications, 2018, vol. 466, no 1, p. 655-675. - https://arxiv.org/pdf/1802.02772.pdf
