arXiv subjects

M. Vaughan

Publications and source records attributed to M. Vaughan.

6 recordsLinked to original sources

arXiv2024

Interior Schauder estimates for fractional elliptic equations in nondivergence form

We obtain sharp interior Schauder estimates for solutions to nonlocal Poisson problems driven by fractional powers of nondivergence form elliptic operators $(-a^{ij}(x) \partial_{ij})^s$, for $0<s<1$, in bounded domains under minimal regularity assumptions on the coefficients $a^{ij}(x)$. Solutions to the fractional problem are characterized by a local degenerate/singular extension problem. We introduce a novel notion of viscosity solutions for the extension problem and implement Caffarelli's perturbation methodology in the corresponding degenerate/singular Monge--Amp\`ere geometry to prove Schauder estimates in the extension. This in turn implies interior Schauder estimates for solutions to the fractional nonlocal equation. Furthermore, we prove a new Hopf lemma, the interior Harnack inequality and H\"older regularity in the Monge--Amp\`ere geometry for viscosity solutions to the extension problem.

math.AP↗

arXiv2023

Continuous symmetrizations and uniqueness of solutions to nonlocal equations

We show that nonlocal seminorms are strictly decreasing under the continuous Steiner rearrangement. This implies that all solutions to nonlocal equations which arise as critical points of nonlocal energies are radially symmetric and decreasing. Moreover, we show uniqueness of solutions by exploiting the convexity of the energies under a tailored interpolation in the space of radially symmetric and decreasing functions. As an application, we consider the long time dynamics of a higher order nonlocal equation which models the growth of symmetric cracks in an elastic medium.

math.AP↗

arXiv2021

Fractional elliptic equations in nondivergence form: definition, applications and Harnack inequality

We define the fractional powers $L^s=(-a^{ij}(x)\partial_{ij})^s$, $0 < s < 1$, of nondivergence form elliptic operators $L=-a^{ij}(x)\partial_{ij}$ in bounded domains $Ω\subset\mathbb{R}^n$, under minimal regularity assumptions on the coefficients $a^{ij}(x)$ and on the boundary $\partialΩ$. We show that these fractional operators appear in several applications such as fractional Monge--Ampère equations, elasticity, and finance. The solution $u$ to the nonlocal Poisson problem $$\begin{cases} (-a^{ij}(x) \partial_{ij})^su = f&\hbox{in}~Ω\\ u=0&\hbox{on}~\partialΩ\end{cases}$$ is characterized by a local degenerate/singular extension problem. We develop the method of sliding paraboloids in the Monge--Ampère geometry and prove the interior Harnack inequality and Hölder estimates for solutions to the extension problem when the coefficients $a^{ij}(x)$ are bounded, measurable functions. This in turn implies the interior Harnack inequality and Hölder estimates for solutions $u$ to the fractional problem.

math.AP↗

arXiv2020

Spin-valve Josephson junctions with perpendicular magnetic anisotropy for cryogenic memory

We demonstrate a Josephson junction with a weak link containing two ferromagnets, with perpendicular magnetic anisotropy and independent switching fields in which the critical current can be set by the mutual orientation of the two layers. Such pseudospin-valve Josephson junctions are a candidate cryogenic memory in an all superconducting computational scheme. Here, we use Pt/Co/Pt/CoB/Pt as the weak link of the junction with $d_\text{Co} = 0.6$ nm, $d_\text{CoB} = 0.3$ nm, and $d_\text{Pt} = 5$ nm and obtain a $60\%$ change in the critical current for the two magnetization configurations of the pseudospin-valve. Ferromagnets with perpendicular magnetic anisotropy have advantages over magnetization in-plane systems which have been exclusively considered to this point, as in principle the magnetization and magnetic switching of layers in the junction should not affect the in-plane magnetic flux.

cond-mat.supr-con↗

arXiv2019

One-sided fractional derivatives, fractional Laplacians, and weighted Sobolev spaces

We characterize one-sided weighted Sobolev spaces $W^{1,p}(\mathbb{R},ω)$, where $ω$ is a one-sided Sawyer weight, in terms of a.e.~and weighted $L^p$ limits as $α\to1^-$ of Marchaud fractional derivatives of order $α$. Similar results for weighted Sobolev spaces $W^{2,p}(\mathbb{R}^n,ν)$, where $ν$ is an $A_p$-Muckenhoupt weight, are proved in terms of limits as $s\to1^-$ of fractional Laplacians $(-Δ)^s$. These are Bourgain--Brezis--Mironescu-type characterizations for weighted Sobolev spaces. We also complement their work by studying a.e.~and weighted $L^p$ limits as $α,s\to0^+$.

math.CA↗

arXiv2018

The Dual Kaczmarz Algorithm

The Kaczmarz algorithm is an iterative method for solving a system of linear equations. It can be extended so as to reconstruct a vector $x$ in a (separable) Hilbert space from the inner-products $\{\langle x, ϕ_{n} \rangle\}$. The Kaczmarz algorithms defines a sequence of approximations from the sequence $\{\langle x, ϕ_{n} \rangle\}$; these approximations only converge to $x$ when $\{ϕ_{n}\}$ is ${effective}$. We dualize the Kaczmarz algorithm so that $x$ can be obtained from $\{\langle x, ϕ_{n} \rangle\}$ by using a second sequence $\{ψ_{n}\}$ in the reconstruction. This allows for the recovery of $x$ even when the sequence $\{ϕ_{n}\}$ is not effective; in particular, our dualization yields a reconstruction when the sequence $\{ϕ_{n}\}$ is $almost$ $effective$. We also obtain some partial results characterizing when the sequence of approximations from $\{\langle x, ϕ_{n} \rangle\}$ using $\{ψ_{n}\}$ converges to $x$, in which case $\{(ϕ_n, ψ_n)\}$ is called an $effective$ $pair$.

math.FA↗