arXiv subjects

Liam Mazurowski

Publications and source records attributed to Liam Mazurowski.

23 records · Page 2Linked to original sources

arXiv2022

Prescribed Mean Curvature Min-Max Theory in Some Non-Compact Manifolds

This paper develops a technique for applying one-parameter prescribed mean curvature min-max theory in certain non-compact manifolds. We give two main applications. First, fix a dimension $3\le n+1 \le 7$ and consider a smooth function $h\colon \mathbb{R}^{n+1}\to \mathbb{R}$ which is asymptotic to a positive constant near infinity. We show that, under certain additional assumptions on $h$, there exists a closed hypersurface $Σ$ in $\mathbb{R}^{n+1}$ with mean curvature prescribed by $h$. Second, let $(M^3,g)$ be an asymptotically flat 3-manifold and fix a constant $c > 0$. We show that, under an additional assumption on $M$, it is possible to find a closed surface $Σ$ of constant mean curvature $c$ in $M$.

math.DG↗

arXiv2020

Cmc Doublings of Minimal Surfaces via Min-Max

Let $Σ^2 \subset M^3$ be a minimal surface of index 0 or 1. Assume that a neighborhood of $Σ$ can be foliated by constant mean curvature (cmc) hypersurfaces. We use min-max theory and the catenoid estimate to construct $\varepsilon$-cmc doublings of $Σ$ for small $\varepsilon > 0$. Such cmc doublings were previously constructed for minimal hypersurfaces $Σ^n \subset M^{n+1}$ with $n+1\ge 4$ by Pacard and Sun using gluing methods.

math.DG↗

arXiv2020

Initial and Boundary Value Problems for the Caputo Fractional Self-Adjoint Difference Equations

In this paper we develop the theory of initial and boundary value problems for the self-adjoint nabla fractional difference equation containing a Caputo fractional nabla difference that is given by \[ \nabla[p(t+1)\nabla_{a*}^νx(t+1)] + q(t)x(t) = h(t), \] where $0 < ν\leq 1$. We give an introduction to the nabla fractional calculus with Caputo fractional differences. We investigate properties of the specific self-adjoint nabla fractional difference equation given above. We prove existence and uniqueness theorems for both initial and boundary value problems under appropriate conditions. We introduce the definition of a Cauchy function which allows us to give a variation of constants formula for solving initial value problems. We then show that this Cauchy function is important in finding a Green's function for a boundary value problem with Sturm-Liouville type boundary conditions. Several inequalities concerning a certain Green's function are derived. These results are important in using fixed point theorems for proving the existence of solutions to boundary value problems for nonlinear fractional equations related to our linear self-adjoint equation.

math.CA↗

arXiv2019

A Weyl law for the $p$-Laplacian

We show that a Weyl law holds for the variational spectrum of the $p$-Laplacian. More precisely, let $(λ_i)_{i=1}^\infty$ be the variational spectrum of $Δ_p$ on a closed Riemannian manifold $(X,g)$ and let $N(λ) = \#\{i:\, λ_i < λ\}$ be the associated counting function. Then we have a Weyl law $N(λ) \sim c \operatorname{vol}(X) λ^{n/p}$. This confirms a conjecture of Friedlander. The proof is based on ideas of Gromov and Liokumovich, Marques, Neves.

math.SP↗

arXiv2016

Generic Extensions and Generic Polynomials for Linear Algebraic Groups

We show the existence of and explicitly construct generic polynomials for various groups, over fields of positive characteristic. The methods we develop apply to a broad class of connected linear algebraic groups defined over finite fields satisfying certain conditions on cohomology. In particular, we use our techniques to study constructions for unipotent groups, certain algebraic tori, and certain split semisimple groups. An attractive consequence of our work is the construction of generic polynomials in the optimal number of parameters for all cyclic 2-groups over all fields of positive characteristic. This contrasts with a theorem of Lenstra, which states no cyclic 2-group of order $\ge 8$ has a generic polynomial over $\mathbb{Q}$.

math.NT↗