Heat kernel estimates for elliptic operators with Robin boundary conditions

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

Name thesis:degree_name

PhD

Level thesis:degree_level

Doctoral

Discipline thesis:degree_discipline

Mathematics

Grantor dc:publisher

ResearchSpace@Auckland

Year dc:date.issued

2021

Author dc:creator

• Wong, Chris

Advisor dc:contributor.advisor

• ter Elst, Tom

Statement dc:rights

• Items in ResearchSpace are protected by copyright, with all rights reserved, unless otherwise indicated.

Licence dc:rights.uri

• https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm

Handle dc:identifier.uri

https://hdl.handle.net/2292/55107

OAI identifier oai:identifier

oai:researchspace.auckland.ac.nz:2292/55107