arXiv subjects

Sullivan Francis MacDonald

Publications and source records attributed to Sullivan Francis MacDonald.

3 recordsLinked to original sources

arXiv2023

Regularity of Singular Solutions to $p$-Poisson Equations

This work showcases level set estimates for weak solutions to the $p$-Poisson equation on a bounded domain, which we use to establish Lebesgue space inclusions for weak solutions. In particular we show that if $Ω\subset\mathbb{R}^n$ is a bounded domain and $u$ is a weak solution to the Dirichlet problem for Poisson's equation \[ -Δu=f\textrm{ in }Ω\] \[ \quad\;\; u=0\textrm{ on }\partialΩ\] for $f\in L^q(Ω)$ with $q<\frac{n}{2}$, then $u\in L^r(Ω)$ for every $r<\frac{qn}{n-2q}$ and indeed $\|u\|_r\leq C\|f\|_q$. This result is shown to be sharp, and similar regularity is established for solutions to the $p$-Poisson equation including in the edge case $q=\frac{n}{p}$.

math.AP↗

arXiv2023

Sum of Squares Decompositions in Hölder Spaces

We investigate the number of half-regular squares required to decompose a non-negative $C^{k,α}(\mathbb{R}^n)$ function into a sum of squares. Each non-negative $C^{3,1}(\mathbb{R}^n)$ function is known to be a finite SOS in $C^{1,1}(\mathbb{R}^n)$, and similar regularity-preserving SOS decompositions have been studied by various authors. Our work refines existing techniques to unify and build upon several known decomposition results, and moreover we provide upper and lower estimates on the number of squares required for SOS decompositions in $C^{k,α}(\mathbb{R}^n)$.

math.FA↗

arXiv2023

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 $Ω\Subset\mathbb{R}^n$, a weight function $v\in L^1_\textrm{loc}(Ω,dx)$, and a suitable non-negative function $τ$, 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)+τ\left|u\right|^{p-2}u&=&f&\textrm{in }Ω, \end{array} \end{align*} \begin{align*} \begin{array}{rccl} u&= & 0&\textrm{on }\partialΩ \end{array} \end{align*} to be bounded and exponentially integrable when the data function $f$ belongs to an appropriate Orlicz space.

math.AP↗