dc.contributor.author Eceizabarrena, D. dc.contributor.author Ponce Vanegas, F. dc.date.accessioned 2021-08-24T15:45:44Z dc.date.available 2021-08-24T15:45:44Z dc.date.issued 2021-08-24 dc.identifier.uri http://hdl.handle.net/20.500.11824/1322 dc.description.abstract We study the problem of pointwise convergence for equations of the type en_US $i\hbar\partial_tu + P(D)u = 0$, 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. dc.format application/pdf en_US dc.language.iso eng en_US dc.rights Reconocimiento-NoComercial-CompartirIgual 3.0 España en_US dc.rights.uri http://creativecommons.org/licenses/by-nc-sa/3.0/es/ en_US dc.subject Pointwise convergence en_US dc.subject dispersive PDEs en_US dc.subject Bourgain's counterexample en_US dc.subject Hausdorff dimension en_US dc.subject mass transference principle en_US dc.title Pointwise Convergence over Fractals for Dispersive Equations with Homogeneous Symbol en_US dc.type info:eu-repo/semantics/article en_US dc.relation.projectID info:eu-repo/grantAgreement/EC/H2020/669689 en_US dc.relation.projectID ES/1PE/SEV-2017-0718 en_US dc.relation.projectID ES/2PE/PGC2018-094528-B-I00 en_US dc.relation.projectID EUS/BERC/BERC.2018-2021 en_US dc.rights.accessRights info:eu-repo/semantics/openAccess en_US dc.type.hasVersion info:eu-repo/semantics/submittedVersion en_US
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