Turner, Andrew James (2021). Quadratic estimates and functional calculi for inhomogeneous first-order operators and applications to boundary value problems for Schrödinger equations. University of Birmingham. Ph.D.
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Turner2021PhD.pdf
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Abstract
We develop a holomorphic functional calculus for first-order operators DB to solve boundary value problems for Schrödinger equations −div A∇u + aVu = 0 in the upper half-space \(ℝ^{n+1}_+\) when n ≥ 3. This relies on quadratic estimates for DB, which are proved for coefficients A, a, V that are independent of the transversal direction to the boundary, and comprised of a complex-elliptic pair A, a that are bounded and measurable, and a singular potential V in the reverse Hölder class \(B^{\frac{n}{2}} (ℝ^n)\). The square function bounds are also shown to be equivalent to non-tangential maximal function bounds. This allows us to prove that the Dirichlet regularity and Neumann boundary value problems with \(L^2( ℝ^n)\)-data are well-posed if and only if certain boundary trace operators defined by the functional calculus are isomorphisms. We prove this property when the coefficient matrices A and a are either a Hermitian or block structure. More generally, the set of all complex-elliptic A for which the boundary value problems are well-posed is shown to be open in \(L^∞\). We also prove these solutions coincide with those generated from the Lax–Milgram Theorem. Furthermore, we extend this theory to prove quadratic estimates for the magnetic Schrödinger operator (∇ + ib)∗A(∇ + ib) when the magnetic field curl (b) is in the reverse Hölder class \(B^{\frac{n}{2}} (ℝ^n)\).
Type of Work: | Thesis (Doctorates > Ph.D.) | |||||||||
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Award Type: | Doctorates > Ph.D. | |||||||||
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Licence: | All rights reserved | |||||||||
College/Faculty: | Colleges (2008 onwards) > College of Engineering & Physical Sciences | |||||||||
School or Department: | School of Mathematics | |||||||||
Funders: | Engineering and Physical Sciences Research Council | |||||||||
Subjects: | Q Science > QA Mathematics | |||||||||
URI: | http://etheses.bham.ac.uk/id/eprint/11585 |
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