Turner, Andrew James (2021). Quadratic estimates and functional calculi for inhomogeneous firstorder operators and applications to boundary value problems for Schrödinger equations. University of Birmingham. Ph.D.

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Abstract
We develop a holomorphic functional calculus for firstorder operators DB to solve boundary value problems for Schrödinger equations −div A∇u + aVu = 0 in the upper halfspace \(ℝ^{n+1}_+\) 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 complexelliptic 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 nontangential maximal function bounds. This allows us to prove that the Dirichlet regularity and Neumann boundary value problems with \(L^2( ℝ^n)\)data are wellposed 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 complexelliptic A for which the boundary value problems are wellposed 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.)  

Award Type:  Doctorates > Ph.D.  
Supervisor(s): 


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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