arXiv subjects

Blair Davey

Publications and source records attributed to Blair Davey.

25 records · Page 2Linked to original sources

arXiv2017

Quantitative uniqueness of solutions to second order elliptic equations with singular potentials in two dimensions

In this article, we study the vanishing order of solutions to second order elliptic equations with singular lower order terms in the plane. In particular, we derive lower bounds for solutions on arbitrarily small balls in terms of the Lebesgue norms of the lower order terms for all admissible exponents. Then we show that a scaling argument allows us to pass from these vanishing order estimates to estimates for the rate of decay of solutions at infinity. Our proofs rely on a new $L^p - L^q$ Carleman estimate for the Laplacian in $\mathbb{R}^2$.

math.AP↗

arXiv2016

Fundamental matrices and Green matrices for non-homogeneous elliptic systems

In this paper, we establish existence, uniqueness, and scale-invariant estimates for fundamental solutions of non-homogeneous second order elliptic systems with bounded measurable coefficients in $\mathbb{R}^n$ and for the corresponding Green functions in arbitrary open sets. We impose certain non-homogeneous versions of de Giorgi-Nash-Moser bounds on the weak solutions and investigate in detail the assumptions on the lower order terms sufficient to guarantee such conditions.

math.AP↗

arXiv2015

The Landis Conjecture for variable coefficient second-order elliptic PDES

In this work, we study the Landis conjecture for second-order elliptic equations in the plane. Precisely, assume that $V\ge 0$ is a measurable real-valued function satisfying $\|V\|_{L^\infty({\mathbb R}^2)} \le 1$. Let $u$ be a real solution to $\mbox{div}(A \nabla u) - V u = 0$ in ${\mathbb R}^2$. Assume that $|u(z)| \le \exp(c_0 |z|)$ and $u(0) = 1$. Then, for any $R$ sufficiently large, \[ \inf_{|z_0| = R} \|u\|_{L^\infty(B_1(z_0))} \ge \exp(- C R \log R). \] In addition to equations with electric potentials, we also derive similar estimates for equations with magnetic potentials. The proofs rely on transforming the equations to Beltrami systems and Hadamard's three-quasi-circle theorem.

math.AP↗

arXiv2015

A further quantification of the unique continuation properties of eigenfunctions of the magnetic Schrödinger operator

We prove quantitative unique continuation results for solutions of $Δw - k^2 w = V w + W\cdot \nabla w$ in a neighborhood of infinity, where $k > 0$, and $V$ and $W$ are complex-valued decaying potentials that satisfy $|V(x)| \lesssim |x|^{-N}$ and $|W(x)| \lesssim |x|^{-P}$ for some $N, P > 1$. For $M(R, 4n/k) = \inf \{||w||_{L^2(B_{4n/k}(x_0))} : |x_0| = R \}$, we show that if the solution $w$ is non-zero, bounded, and normalized, then $M(R, 4n/k) \gtrsim \exp(-kR - G \log R)$, where $G > \frac{n-1}{2}$ is a constant. An examination of radial solutions to $Δw - k^2 w = V w + W\cdot \nabla w$ shows that this new estimate for $M(R, 4n/k)$ is sharp up to logarithmic terms.

math.AP↗

arXiv2014

Sharp constructions of eigenfunctions of the magnetic Schrödinger operator

We prove sharpness of quantitative unique continuation results for solutions of $-Δu + W\cdot \nabla u + V u = \la u$, where $\la \in \C$ and $V$ and $W$ are complex-valued decaying potentials that satisfy $|V(x)| \lesssim ^{-N}$ and $|W(x)| \lesssim ^{-P}$. For $M(R) = \inf_{|x_0| = R}||u||_{L^2(B_1(x_0))}$, it was shown in a companion paper that if the solution $u$ is non-zero, bounded, and $u(0) = 1$, then $M(R) \gtrsim \exp(-C R^{\be_0}(\log R)^{A(R)})$, where $\be_0 = max{2 - 2P, (4-2N)/3, 1}$. Under certain conditions on $N$, $P$, $\la$, and the dimension, we construct examples (some of which are in the style of Meshkov) to prove that this estimate for $M(R)$ is sharp. That is, we construct functions $u$, $V$ and $W$ such that $-Δu + W\cdot \nabla u + V u = \la u$, $|V(x)| \lesssim ^{-N},$ $|W(x)| \lesssim ^{-P}$ and $|u(x)| \lesssim \exp(-c|x|^{\be_0}(\log |x|)^C)$.

math.AP↗

arXiv2014

Some quantitative unique continuation results for eigenfunctions of the magnetic Schrödinger operator

We prove quantitative unique continuation results for solutions of $-Δu + W\cdot \nabla u + Vu = λu$, where $λ\in \mathbb{C}$ and $V$ and $W$ are complex-valued decaying potentials that satisfy $|V(x)| \lesssim \langle x\rangle^{-N}$ and $|W(x)| \lesssim \langle x\rangle^{-P}$. For $M(R) = \inf_{|x_0| = R}||u||_{L^2(B_1(x_0))}$, we show that if the solution $u$ is non-zero, bounded, and $u(0) = 1$, then $M(R) \gtrsim \exp(-C R^{β_0}(\log R)^{A( R)})$, where $β_0 = \max\{2 - 2P, \frac{4-2N}{3}, 1\}$. Under certain conditions on $N$, $P$ and $λ$, we construct examples (some of which are in the style of Meshkov) to prove that this estimate for $M(R)$ is sharp. That is, we construct functions $u, V$ and $W$ such that $-Δu + W\cdot \nabla u + Vu = λu$, $|V(x)| \lesssim \langle x\rangle^{-N}$, $|W(x)| \lesssim \langle x\rangle^{-P}$ and $|u(x)| \lesssim \exp(-c|x|^{β_0}(\log |x|)^C)$.

math.AP↗

arXiv2014

A Meshkov-type construction for the borderline case

We construct functions $u: \mathbb{R}^2 \to \mathbb{C}$ that satisfy an elliptic eigenvalue equation of the form $-Δu + W \cdot \nabla u + V u = λu$, where $λ\in \mathbb{C}$, and $V$ and $W$ satisfy $|V(x)| \lesssim ^{-N}$, and $|W(x)| \lesssim ^{-P}$, with $\min\{N, P\} = 1/2$. For $|x|$ sufficiently large, these solutions satisfy $|u(x)| \lesssim \exp(- c|x|)$. In the author's previous work, examples of solutions over $\mathbb{R}^2$ were constructed for all $N, P$ such that $\min\{N,P\} \in [0, 1/2)$. These solutions were shown to have the optimal rate of decay at infinity. A recent result of Lin and Wang shows that the constructions presented in this note for the borderline case of $\min\{N, P\} = 1/2$ also have the optimal rate of decay at infinity.

math.AP↗