Browsing by Author "Davey, Blair"

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Improved quantitative unique continuation for complex-valued drift equations in the plane
(Walter de Gruyter GmbH, 2022-11) Davey, Blair; Kenig, Carlos; Wang, Jenn-Nan
In this article, we investigate the quantitative unique continuation properties of complex-valued solutions to drift equations in the plane. We consider equations of the form Δu+W⋅∇u=0 in R2 , where W=W1+iW2 with each Wj being real-valued. Under the assumptions that Wj∈Lqj for some q1∈[2,∞] , q2∈(2,∞] and that W2 exhibits rapid decay at infinity, we prove new global unique continuation estimates. This improvement is accomplished by reducing our equations to vector-valued Beltrami systems. Our results rely on a novel order of vanishing estimate combined with a finite iteration scheme.
A Quantification of a Besicovitch Non-linear Projection Theorem via Multiscale Analysis
(Springer Science and Business Media LLC, 2022-02) Davey, Blair; Taylor, Krystal
The Besicovitch projection theorem states that if a subset E of the plane has finite length in the sense of Hausdorff measure and is purely unrectifiable (so its intersection with any Lipschitz graph has zero length), then almost every orthogonal projection of E to a line will have zero measure. In other words, the Favard length of a purely unrectifiable 1-set vanishes. In this article, we show that when linear projections are replaced by certain non-linear projections called curve projections, this result remains true. In fact, we go further and use multiscale analysis to prove a quantitative version of this Besicovitch non-linear projection theorem. Roughly speaking, we show that if a subset of the plane has finite length in the sense of Hausdorff and is nearly purely unrectifiable, then its Favard curve length is very small. Our techniques build on those of Tao, who in (Proc Lond Math Soc 98:559–584, 2009) proves a quantification of the original Besicovitch projection theorem.
Variable-coefficient parabolic theory as a high-dimensional limit of elliptic theory
(Springer Science and Business Media LLC, 2024-01) Davey, Blair; Vega Garcia, Mariana Smit
This paper continues the study initiated in Davey (Arch Ration Mech Anal 228:159–196, 2018), where a high-dimensional limiting technique was developed and used to prove certain parabolic theorems from their elliptic counterparts. In this article, we extend these ideas to the variable-coefficient setting. This generalized technique is demonstrated through new proofs of three important theorems for variable-coefficient heat operators, one of which establishes a result that is, to the best of our knowledge, also new. Specifically, we give new proofs of L2 → L2 Carleman estimates and the monotonicity of Almgren-type frequency functions, and we prove a new monotonicity of Alt–Caffarelli–Friedman-type functions. The proofs in this article rely only on their related elliptic theorems and a limiting argument. That is, each parabolic theorem is proved by taking a high-dimensional limit of a related elliptic result.