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DC Field | Value | Language |
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dc.contributor.advisor | Polidoro, Sergio | - |
dc.contributor.author | Rebucci, Annalaura | - |
dc.date.accessioned | 2023-06-22T12:32:13Z | - |
dc.date.available | 2023-06-22T12:32:13Z | - |
dc.date.issued | 2023 | - |
dc.identifier.uri | https://hdl.handle.net/1889/5388 | - |
dc.description.abstract | In this thesis, we are concerned with the regularity theory of strongly degenerate Kolmogorov equations and we also study a relativistic generalization of such equations. We divide this dissertation into three parts. In the first part, we present some results which lie in the classical regularity theory of Kolmogorov-type operators with regular coefficients. In particular, we here discuss some Schauder estimates for classical solutions to Kolmogorov equations in non-divergence form with Dini-continuous coefficients and right-hand side. Furthermore, we show new pointwise regularity results and a Taylor-type expansion up to second order with estimate of the rest in L^p norm. The second part focuses on the weak regularity theory of degenerate Kolmogorov equations with discontinuous coefficients, which is nowadays the main focus of the research community. More precisely, we present a Harnack inequality and the Hölder continuity for weak solutions to the Kolmogorov equation with measurable coefficients, integrable lower order terms and nonzero source term. We subsequently prove the existence of a fundamental solution associated to the Kolmogorov operator, together with Gaussian lower and upper bounds. Finally, in the last part of this thesis, we address a possible generalization of the kinetic Kolmogorov-Fokker-Planck equation, which is in accordance with the theory of special relativity. In particular, we explain why the operator proposed is the suitable relativistic generalization of the Fokker-Planck operator and we describe it as a Hörmander operator which is invariant with respect to Lorentz transformations. | en_US |
dc.language.iso | Inglese | en_US |
dc.publisher | Università degli studi di Parma. Dipartimento di Scienze matematiche, fisiche e informatiche | en_US |
dc.relation.ispartofseries | Dottorato di ricerca in Matematica | en_US |
dc.rights | © Annalaura Rebucci, 2023 | en_US |
dc.rights.uri | http://creativecommons.org/licenses/by-nc/4.0/ | * |
dc.subject | Degenerate Kolmogorov equations | en_US |
dc.subject | Regularity theory | en_US |
dc.subject | Classical solutions | en_US |
dc.subject | Taylor formula | en_US |
dc.subject | Dini regularity | en_US |
dc.subject | Pointwise regularity | en_US |
dc.subject | BMO pointwise estimate | en_US |
dc.subject | VMO pointwise estimate | en_US |
dc.subject | Weak regularity theory | en_US |
dc.subject | Weak Poincaré inequality | en_US |
dc.subject | Harnack inequality | en_US |
dc.subject | Hölder regularity | en_US |
dc.subject | Ultraparabolic | en_US |
dc.subject | Fundamental solution | en_US |
dc.subject | Potential theory | en_US |
dc.subject | Asymptotic bounds | en_US |
dc.subject | Special relativity | en_US |
dc.title | Regularity results and new perspectives for degenerate Kolmogorov equations | en_US |
dc.type | Doctoral thesis | en_US |
dc.subject.miur | MAT/05 | en_US |
dc.rights.license | Attribuzione - Non commerciale 4.0 Internazionale | * |
dc.rights.license | Attribuzione - Non commerciale 4.0 Internazionale | * |
dc.rights.license | Attribuzione - Non commerciale 4.0 Internazionale | * |
Appears in Collections: | Matematica. Tesi di dottorato |
Files in This Item:
File | Description | Size | Format | |
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Relazione attività svolta.pdf Restricted Access | Relazione attività svolta | 86.39 kB | Adobe PDF | View/Open Request a copy |
Thesis_Rebucci.pdf | Thesis | 1.51 MB | Adobe PDF | View/Open |
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