Heat kernel estimates for elliptic operators with Robin boundary conditions

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dc.contributor.advisor ter Elst, Tom
dc.contributor.author Wong, Chris
dc.date.accessioned 2021-05-17T21:12:46Z
dc.date.available 2021-05-17T21:12:46Z
dc.date.issued 2021 en
dc.identifier.uri https://hdl.handle.net/2292/55107
dc.description.abstract Consider the elliptic operator on a bounded connected open set Ω ⊂ Rd where d ≥ 2, subject to Robin boundary conditions ∂νu + βu = 0. We show that the kernel for the semigroup generated by −A satisfies Gaussian and Hölder Gaussian bounds given domain and coefficients regularities. In particular we show that when the domain is Lipschitz and the principal coefficients are real, then the kernel is ν-Hölder continuous for some ν ∈ (0, 1). We also show that if the domain is C 1+κ , where κ ∈ (0, 1), and the coefficients are κ-Hölder continuous, then the kernel is differentiable and the derivative is κ-Hölder continuous. We use these kernel estimates to prove other properties of the semigroup, including holomorphy and irreducibility. Moreover, we prove lower bounds for the kernel if the domain is Lipschitz, all coefficients are real and A is self-adjoint. As an application we also associate the elliptic operator with the Dirichlet-to-Neumann operator N . We show that if Ω is C 1+κ , where κ ∈ (0, 1), ckl = clk are real κ-Hölder continuous, ak = bk = 0 and a0 is real, then the kernel of the semigroup generated by −N has a Hölder Poisson bounds
dc.publisher ResearchSpace@Auckland en
dc.relation.ispartof PhD Thesis - University of Auckland en
dc.relation.isreferencedby UoA en
dc.rights Items in ResearchSpace are protected by copyright, with all rights reserved, unless otherwise indicated. en
dc.rights Items in ResearchSpace are protected by copyright, with all rights reserved, unless otherwise indicated.
dc.rights.uri https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm en
dc.rights.uri http://creativecommons.org/licenses/by-nc-sa/3.0/nz/
dc.title Heat kernel estimates for elliptic operators with Robin boundary conditions
dc.type Thesis en
thesis.degree.discipline Mathematics
thesis.degree.grantor The University of Auckland en
thesis.degree.level Doctoral en
thesis.degree.name PhD en
dc.date.updated 2021-05-11T04:39:41Z
dc.rights.holder Copyright: The author en
dc.rights.accessrights http://purl.org/eprint/accessRights/OpenAccess en


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