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Scott Rodney

Publications and source records attributed to Scott Rodney.

At least 19 recordsLinked to original sources
arXiv2026

Degenerate Sobolev and Poincar\'e inequalities via extrapolation

In this paper we prove matrix weighted Sobolev and Poincar\'e inequalities using techniques derived from the theory of Rubio de Francia extrapolation. Given weights $w,\,v$ and a symmetric non-negative definite matrix valued function $Q$ defined on a connected open subset $\Omega$ of $\mathbb{f}R^n$ that satisfies the lower ellipticity condition \[ w(x)^p \leq |\sqrt{Q(x)}\xi|^p,\quad \xi\in \mathbb{R}^n, \] we give Lebesgue integrability conditions on the weights $w,v$ that ensure there exists $\tau\geq 1$ so that Sobolev and Poincar\'e inequalities of the form \[\bigg(\int_\Omega |u|^{\tau p} \,vdx\bigg)^{\frac{1}{\tau p}} \leq C(v,w) \bigg(\int_\Omega |\sqrt{Q}\nabla u|^p\,dx\bigg)^{\frac{1}{ p}},\textrm{ and}\] \[\bigg(\int_\Omega |u-\langle u\rangle_{\Omega,v}|^{\tau p} \,v dx\bigg)^\frac{1}{\tau p} \leq C(v,w)\bigg(\int_\Omega|\sqrt{Q}\nabla u|^p \, dx\bigg)^{\frac{1}{p}}\] hold for smooth $u$. We explore these and related results in the context of several examples that include John domains, the Heisenberg group, and CR manifolds.

math.AP↗
arXiv2026

On $H=W$ in Banach function spaces

In this paper we prove ``$H=W$" in the context of a Banach function space $X(\Omega)$. Let $\Omega$ be a subset of ${\mathbb R}^n$ and denote by $W^1_X(\Omega)$ the collection of all those $f\in X(\Omega)$ whose distributional derivatives $\partial_jf$ are contained in $X(\Omega)$. Our main result provides a small collection of ``universal" hypotheses on $X(\Omega)$ that ensure $W^1_X(\Omega)$ is equal to $H^1_X(\Omega)$, the formal closure of ${Lip}(\Omega)\cap W^1_X(\Omega)$ with respect to the norm \[\|f\|_{W^1_X(\Omega)} = \|f\|_{X(\Omega)} + \|\nabla f\|_{X(\Omega)}.\] The main theorem has two corollaries. The first gives a slightly stronger set of hypotheses for ``$H=W$", and the second gives density of $C^\infty_c({\mathbb R}^n)$ in $W^1_X({\mathbb R}^n)$.

math.FA↗
arXiv2025

Existence and uniqueness of solutions of degenerate elliptic equations with lower order terms

We prove the existence and uniqueness of solutions to a Dirichlet problem \[ \begin{cases} Lu = f + v^{-1}\text{Div}(v{\bf e} h), & x \in \Omega; u = 0, & x \in \partial \Omega, \end{cases}\] where $L$ is a degenerate, linear, second order elliptic operator with lower order terms. We assume very weak hypotheses, in terms of the coefficients of the equation, and we also assume the existence of degenerate Sobolev and Poincar\'e inequalities. One notable feature of our result is that we show that we can assume significantly weaker versions of the Sobolev inequality if we in turn assume stronger integrability conditions on the coefficients. Our theorems generalize a number of results in the literature on degenerate elliptic equations.

math.AP↗
arXiv2025

Geometric Structure in Weighted Alpert Wavelets

In this paper we present a number of results concerning Alpert wavelet bases for $L^2(\mu)$, with $\mu$ a locally finite positive Borel measure on $\mathbb{R}^n$. We show that the properties of such a basis depend on linear dependences in $L^2(\mu)$ among the functions from which the wavelets are constructed; this result completes an investigation begun by Rahm, Sawyer, and Wick in arXiv:1808.01223. We also show that a Gr\"{o}bner basis technique can be used to efficiently detect these dependences. Lastly we give a generalization of the Alpert basis construction, where the amount of orthogonality in the basis is allowed to vary over the dyadic grid.

math.CA↗
arXiv2024

Bounded solutions of degenerate elliptic equations with an Orlicz-gain Sobolev inequality

We consider the boundedness and exponential integrability of solutions to the Dirichlet problem for the degenerate elliptic equation \[ -v^{-1}\mathrm{Div}(|\sqrt{Q}\nabla u|^{p-2}Q\nabla u)=f|f|^{p-2}- v^{-1}\mathrm{Div}(v|g|^{p-2}g \mathbf{t}), \quad 1 1$. In our results we study the interplay between the Sobolev inequality and the regularity assumptions needed on $f$ and $g$ to prove that the solution is bounded or is exponentially integrable. Our results generalize those previously proved in previous work by the authors.

math.AP↗
arXiv2023

Matrix Weights and Regularity for Degenerate Elliptic Equations

We prove local boundedness, Harnack's inequality and local regularity for weak solutions of quasilinear degenerate elliptic equations in divergence form with Rough coefficients. Degeneracy is encoded by a non-negative, symmetric, measurable matrix valued function Q(x) and two suitable non-negative weight functions. We setup an axiomatic approach in terms of suitable geometric conditions and local Sobolev-Poincar\'e inequalities. Data integrability is close to L1 and is exploited in terms of a suitable Stummel-Kato class that in some cases is necessary for local regularity.

math.AP↗
arXiv2022

Bounded Weak Solutions of Degenerate $p$-Poisson Equations

In this work we study global boundedness and exponential integrability of weak solutions to degenerate $p$-Poisson equations using an iterative method of De Giorgi type. Given a symmetric, non-negative definite matrix valued function $Q$ defined on a bounded domain $\Omega\Subset\mathbb{R}^n$, a weight function $v\in L^1_\textrm{loc}(\Omega,dx)$, and a suitable non-negative function $\tau$, we give sufficient conditions for any weak solution to the Dirichlet problem \begin{align*} \begin{array}{rccl} -\displaystyle\frac{1}{v}\mathrm{{div}}\left(\left|\sqrt{Q}\nabla u\right|^{p-2}Q\nabla u\right)+\tau\left|u\right|^{p-2}u&=&f&\textrm{in }\Omega, \end{array} \end{align*} \begin{align*} \begin{array}{rccl} u&= & 0&\textrm{on }\partial\Omega \end{array} \end{align*} to be bounded and exponentially integrable when the data function $f$ belongs to an appropriate Orlicz space.

math.AP↗
arXiv2022

Families of Young Functions and Limits of Orlicz Norms

Given a $\sigma$-finite measure space $(X,\mu)$, a Young function $\Phi$, and a one-parameter family of Young functions $\{\Psi_q\}$, we find necessary and sufficient conditions for the associated Orlicz norms of any function $f\in L^\Phi(X,\mu)$ to satisfy \[ \lim_{q\rightarrow \infty}\|f\|_{L^{\Psi_q}(X,\mu)}=C\|f\|_{L^\infty(X,\mu)}. \] The constant $C$ is independent of $f$ and depends only on the family $\{\Psi_q\}$. Several examples of one-parameter families of Young functions satisfying our conditions are given, along with counterexamples when our conditions fail.

math.AP↗
arXiv2021

Poincar\'e Inequalities and Neumann Problems for the Variable Exponent Setting

We extend the results of [5], where we proved an equivalence between weighted Poincar\'e inequalities and the existence of weak solutions to a family of Neumann problems related to a degenerate $p$-Laplacian. Here we prove a similar equivalence between Poincar\'e inequalities in variable exponent spaces and solutions to a degenerate $p(x)$-Laplacian, a non-linear elliptic equation with nonstandard growth conditions.

math.AP↗
arXiv2021

The Cognition of Counterexample in Mathematics Students

Studying Mathematics requires a synthesis of skills from a multitude of academic disciplines; logical reasoning being chief among them. This paper explores mathematical logical preparedness of students entering first year university mathematics courses and also the effectiveness of using logical facility to predict successful course outcomes. We analyze data collected from students enrolled at the University of Winnipeg in a pre-service course for high school teachers. We do find that, being able to successfully answer logical questions, both before and after intervention, are significant in relation to improved student outcomes.

math.HO↗
arXiv2020

A note on the limit of Orlicz norms

We generalize the well-known inequality that the limit of the $L^p$ norm of a function as $p\rightarrow\infty$ is the $L^\infty$ norm to the scale of Orlicz spaces.

math.CA↗
arXiv2020

Bounded weak solutions to elliptic PDE with data in Orlicz spaces

A classical regularity result is that non-negative solutions to the Dirichlet problem $\Delta u =f$ in a bounded domain $\Omega$, where $f\in L^q(\Omega)$, $q>\frac{n}2$, satisfy $\|u\|_{L^\infty(\Omega)} \leq C\|f\|_{L^q(\Omega)}$. We extend this result in three ways: we replace the Laplacian with a degenerate elliptic operator; we show that we can take the data $f$ in an Orlicz space $L^A(\Omega)$ that lies strictly between $L^{\frac{n}{2}}(\Omega)$ and $L^q(\Omega)$, $q>\frac{n}2$; and we show that that we can replace the $L^A$ norm in the right-hand side by a smaller expression involving the logarithm of the "entropy bump" $\|f\|_{L^A(\Omega)}/\|f\|_{L^{\frac{n}{2}}(\Omega)}$, generalizing a result due to Xu.

math.AP↗
arXiv2019

An Improved Compact Embedding Theorem for Degenerate Sobolev Spaces

This short note investigates the compact embedding of degenerate matrix weighted Sobolev spaces into weighted Lebesgue spaces. The Sobolev spaces explored are defined as the abstract completion of Lipschitz functions in a bounded domain $\Omega$ with respect to the norm: $$\|f\|_{QH^{1,p}(v,\mu;\Omega)} = \|f\|_{L^p_v(\Omega)} + \|\nabla f\|_{\mathcal{L}^p_Q(\mu;\Omega)}$$ where the weight $v$ is comparable to a power of the pointwise operator norm of the matrix valued function $Q=Q(x)$ in $\Omega$. Following our main theorem, we give an explicit application where degeneracy is controlled through an ellipticity condition of the form $$w(x)|\xi|^p \leq \left(\xi\cdot Q(x)\xi\right)^{p/2}\leq \tau(x)|\xi|^p$$ for a pair of $p$-admissible weights $w\leq \tau$ in $\Omega$. We also give explicit examples demonstrating the sharpness of our hypotheses.

math.AP↗
arXiv2018

Global Sobolev inequalities and Degenerate P-Laplacian equations

We prove that a local, weak Sobolev inequality implies a global Sobolev estimate using existence and regularity results for a family of $p$-Laplacian equations. Given $\Omega\subset\mathbb{R}^n$, let $\rho$ be a quasi-metric on $\Omega$, and let $Q$ be an $n\times n$ semi-definite matrix function defined on $\Omega$. For an open set $\Theta\Subset\Omega$, we give sufficient conditions to show that if the local weak Sobolev inequality % \[ \Big(\fint_B |f|^{p\sigma}dx\Big)^\frac{1}{p\sigma} \leq C\Big[ r(B)\fint_B |\sqrt{Q}\nabla f|^pdx + \fint_B |f|^pdx\Big]^\frac{1}{p} \] holds for some $\sigma>1$, all balls $B\subset \Theta$, and functions $f\in Lip_0(\Theta)$, then the global Sobolev inequality \[ \Big(\int_\Theta |f|^{p\sigma}dx\Big)^\frac{1}{p\sigma} \leq C\Big(\int_\Theta |\sqrt{Q}\nabla f(x)|^pdx\Big)^\frac{1}{p} \] also holds. Central to our proof is showing the existence and boundedness of solutions of the Dirichlet problem \[ \begin{cases} \mx_{p,\tau} u & = \varphi \text{in} \Theta \\ u & = 0 \text{in} \partial \Theta, \end{cases} \] where $\mx_{p,\tau}$ is a degenerate $p$-Laplacian operator with a zero order term: \[ \mx_{p,\tau} u = \text{div}\Big(\big|\sqrt{Q} \nabla u\big|^{p-2}Q\nabla u\Big) - \tau |u|^{p-2}u. \]

math.AP↗
arXiv2017

Poincare Inequalities and Neumann Problems for the p-Laplacian

We prove an equivalence between weighted Poincare inequalities and the existence of weak solutions to a Neumann problem related to a degenerate p- Laplacian. The Poincare inequalities are formulated in the context of degenerate Sobolev spaces defined in terms of a quadratic form, and the associated matrix is the source of the degeneracy in the p-Laplacian.

math.AP↗
arXiv2015

Matrix $A_p$ weights, degenerate Sobolev spaces, and mappings of finite distortion

We study degenerate Sobolev spaces where the degeneracy is controlled by a matrix $A_p$ weight. This class of weights was introduced by Nazarov, Treil and Volberg, and degenerate Sobolev spaces with matrix weights have been considered by several authors for their applications to PDEs. We prove that the classical Meyers-Serrin theorem, H = W, holds in this setting. As applications we prove partial regularity results for weak solutions of degenerate p-Laplacian equations, and in particular for mappings of finite distortion.

math.AP↗
arXiv2014

Harnack's inequality and H\"older continuity for weak solutions of degenerate quasilinear equations with rough coefficients

We continue to study regularity results for weak solutions of the large class of second order degenerate quasilinear equations of the form \begin{eqnarray} \text{div}\big(A(x,u,\nabla u)\big) = B(x,u,\nabla u)\text{ for }x\in\Omega\nonumber \end{eqnarray} as considered in our previous paper giving local boundedness of weak solutions. Here we derive a version of Harnack's inequality as well as local H\"older continuity for weak solutions. The possible degeneracy of an equation in the class is expressed in terms of a nonnegative definite quadratic form associated with its principal part. No smoothness is required of either the quadratic form or the coefficients of the equation. Our results extend ones obtained by J. Serrin and N. Trudinger for quasilinear equations, as well as ones for subelliptic linear equations obtained by Sawyer and Wheeden in their 2006 AMS memoir article.

math.AP↗