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dc.contributor.authorEceizabarrena, D.
dc.contributor.authorPonce Vanegas, F. 
dc.date.accessioned2021-08-24T15:45:44Z
dc.date.available2021-08-24T15:45:44Z
dc.date.issued2021-08-24
dc.identifier.urihttp://hdl.handle.net/20.500.11824/1322
dc.description.abstractWe study the problem of pointwise convergence for equations of the type $i\hbar\partial_tu + P(D)u = 0$, where the symbol $P$ is real, homogeneous and non-singular. We prove that for initial data $f\in H^s(\mathbb{R}^n)$ with $s>(n-\alpha+1)/2$ the solution $u$ converges to $f$ $\mathcal{H}^\alpha$-a.e, where $\mathcal{H}^\alpha$ is the $\alpha$-dimensional Hausdorff measure. We improve upon this result depending on the dispersive strength of the symbol. On the other hand, we prove negative results for a wide family of polynomial symbols $P$. Given $\alpha$, we exploit a Talbot-like effect to construct regular initial data whose solutions $u$ diverge in sets of Hausdorff dimension $\alpha$. However, for quadratic symbols like the saddle, other kind of examples show that our positive results are sometimes best possible. To compute the dimension of the sets of divergence we use a Mass Transference Principle from Diophantine approximation theory.en_US
dc.formatapplication/pdfen_US
dc.language.isoengen_US
dc.rightsReconocimiento-NoComercial-CompartirIgual 3.0 Españaen_US
dc.rights.urihttp://creativecommons.org/licenses/by-nc-sa/3.0/es/en_US
dc.subjectPointwise convergenceen_US
dc.subjectdispersive PDEsen_US
dc.subjectBourgain's counterexampleen_US
dc.subjectHausdorff dimensionen_US
dc.subjectmass transference principleen_US
dc.titlePointwise Convergence over Fractals for Dispersive Equations with Homogeneous Symbolen_US
dc.typeinfo:eu-repo/semantics/articleen_US
dc.relation.projectIDinfo:eu-repo/grantAgreement/EC/H2020/669689en_US
dc.relation.projectIDES/1PE/SEV-2017-0718en_US
dc.relation.projectIDES/2PE/PGC2018-094528-B-I00en_US
dc.relation.projectIDEUS/BERC/BERC.2018-2021en_US
dc.rights.accessRightsinfo:eu-repo/semantics/openAccessen_US
dc.type.hasVersioninfo:eu-repo/semantics/submittedVersionen_US


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Reconocimiento-NoComercial-CompartirIgual 3.0 España
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