Please use this identifier to cite or link to this item: http://hdl.handle.net/10773/35429
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dc.contributor.authorPlakhov, Alexanderpt_PT
dc.date.accessioned2022-12-14T10:21:42Z-
dc.date.available2022-12-14T10:21:42Z-
dc.date.issued2022-10-
dc.identifier.issn0944-2669pt_PT
dc.identifier.urihttp://hdl.handle.net/10773/35429-
dc.description.abstractLet $u$ minimize the functional $F(u) = \int_\Omega f(\nabla u(x))\, dx$ in the class of convex functions $u : \Omega \to {\mathbb R}$ satisfying $0 \le u \le M$, where $\Omega \subset {\mathbb R}^2$ is a compact convex domain with nonempty interior and $M > 0$, and $f : {\mathbb R}^2 \to {\mathbb R}$ is a $C^2$ function, with $\{ \xi : \, \text{the smallest eigenvalue of} \, f''(\xi) \, \text{is zero} \}$ being a closed nowhere dense set in ${\mathbb R}^2$. Let epi$(u)$ denote the epigraph of $u$. Then any extremal point $(x, u(x))$ of epi$(u)$ is contained in the closure of the set of singular points of epi$(u)$. As a consequence, an optimal function $u$ is uniquely defined by the set of singular points of epi$(u)$. This result is applicable to the classical Newton's problem, where $F(u) = \int_\Omega (1 + |\nabla u(x)|^2)^{-1}\, dx$.pt_PT
dc.language.isoengpt_PT
dc.publisherSpringer Naturept_PT
dc.relationinfo:eu-repo/grantAgreement/FCT/6817 - DCRRNI ID/UIDB%2F04106%2F2020/PTpt_PT
dc.relationinfo:eu-repo/grantAgreement/FCT/6817 - DCRRNI ID/UIDP%2F04106%2F2020/PTpt_PT
dc.rightsrestrictedAccesspt_PT
dc.titleA solution to Newton’s least resistance problem is uniquely defined by its singular setpt_PT
dc.typearticlept_PT
dc.description.versionpublishedpt_PT
dc.peerreviewedyespt_PT
degois.publication.issue5pt_PT
degois.publication.titleCalculus of Variations and Partial Differential Equationspt_PT
degois.publication.volume61pt_PT
dc.relation.publisherversionhttps://doi.org/10.1007/s00526-022-02300-wpt_PT
dc.identifier.doi10.1007/s00526-022-02300-wpt_PT
dc.identifier.essn1432-0835pt_PT
dc.identifier.articlenumber189pt_PT
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