Analysis of nonlocal equations: One-sided weighted fractional Sobolev spaces and Harnack inequality for fractional nondivergence form elliptic equations

dc.contributor.advisor Pablo Raúl Stinga
dc.contributor.author Vaughan, Mary
dc.contributor.department Mathematics
dc.date 2020-09-23T19:12:26.000
dc.date.accessioned 2021-02-25T21:36:56Z
dc.date.available 2021-02-25T21:36:56Z
dc.date.copyright Sat Aug 01 00:00:00 UTC 2020
dc.date.embargo 2020-09-10
dc.date.issued 2020-01-01
dc.description.abstract <p>This dissertation deals with the following two projects.</p> <p>First, we characterize one-sided weighted Sobolev spaces W^{1,p}(R,ω), where ω is a one-sided Sawyer weight, in terms of a.e. and weighted L^p limits as α → 1− of Marchaud fractional derivatives of order 0 < α < 1. These are Bourgain–Brezis–Mironescu-type characterizations for one-sided weighted Sobolev spaces. Similar results for weighted Sobolev spaces W^{2,p}(R^n,ν), where ν is an A_p-Muckenhoupt weight, are proved in terms of limits as s → 1− of fractional Laplacians (−∆)^s. We also additionally study the a.e. and weighted L^p limits as α, s → 0+.</p> <p>Second, we define fractional powers of nondivergence form elliptic operators (−aij(x)∂ij)^s for 0 < s < 1 with Hölder coefficients and characterize a Poisson problem driven by (−aij(x)∂ij)^s with a local degenerate extension problem. An interior Harnack inequality for nonnegative solutions to such an extension equation with bounded, measurable coefficients is proved. This in turn implies the interior Harnack inequality for the fractional problem.</p>
dc.format.mimetype application/pdf
dc.identifier archive/lib.dr.iastate.edu/etd/18241/
dc.identifier.articleid 9248
dc.identifier.contextkey 19236840
dc.identifier.doi https://doi.org/10.31274/etd-20200902-160
dc.identifier.s3bucket isulib-bepress-aws-west
dc.identifier.submissionpath etd/18241
dc.identifier.uri https://dr.lib.iastate.edu/handle/20.500.12876/94393
dc.language.iso en
dc.source.bitstream archive/lib.dr.iastate.edu/etd/18241/Vaughan_iastate_0097E_18949.pdf|||Fri Jan 14 21:39:11 UTC 2022
dc.subject.keywords Fractional derivatives
dc.subject.keywords Harnack inequalities
dc.subject.keywords linearized Monge-Ampère equation
dc.subject.keywords One-sided weights
dc.subject.keywords sliding paraboloids
dc.subject.keywords weighted Sobolev spaces
dc.title Analysis of nonlocal equations: One-sided weighted fractional Sobolev spaces and Harnack inequality for fractional nondivergence form elliptic equations
dc.type article
dc.type.genre dissertation
dspace.entity.type Publication
relation.isOrgUnitOfPublication 82295b2b-0f85-4929-9659-075c93e82c48
thesis.degree.discipline Mathematics
thesis.degree.level dissertation
thesis.degree.name Doctor of Philosophy
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