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Russell M. Brown

Publications and source records attributed to Russell M. Brown.

8 recordsLinked to original sources
arXiv2026

Recovering nonsmooth coefficients for higher-order perturbations of a polyharmonic operator

We consider an inverse problem for a higher order elliptic operator where the principal part is the polyharmonic operator $(-Δ)^m$ with $ m \geq 2$. We show that the map from the coefficients to a certain bilinear form is injective. We have a particular focus on obtaining these results under lower regularity on the coefficients. It is known that knowledge of this bilinear form is equivalent to a knowledge of a Dirichlet to Neumann map or the Cauchy data for solutions.

math.AP↗
arXiv2021

Extendability of functions with partially vanishing trace

Let $Ω\subseteq \mathbb{R}^d$ be open and $D\subseteq \partialΩ$ be a closed part of its boundary. Under very mild assumptions on $Ω$, we construct a bounded Sobolev extension operator for the Sobolev space $\mathrm{W}^{k , p}_D (Ω)$, $1 \leq p < \infty$, which consists of all functions in $\mathrm{W}^{k , p} (Ω)$ that vanish in a suitable sense on $D$. In contrast to earlier work, this construction is global and \emph{not} using a localization argument, which allows to work with a boundary regularity that is sharp at the interface dividing $D$ and $\partial Ω\setminus D$. Moreover, we provide homogeneous and local estimates for the extension operator. Also, we treat the case of Lipschitz function spaces with a vanishing trace condition on $D$.

math.CA↗
arXiv2019

The mixed problem for the Laplacian in Lipschitz domains

We consider the mixed boundary value problem or Zaremba's problem for the Laplacian in a bounded Lipschitz domain in R^n. We specify Dirichlet data on part of the boundary and Neumann data on the remainder of the boundary. We assume that the boundary between the sets where we specify Dirichlet and Neumann data is a Lipschitz surface. We require that the Neumann data is in L^p and the Dirichlet data is in the Sobolev space of functions having one derivative in L^p for some p near 1. Under these conditions, there is a unique solution to the mixed problem with the non-tangential maximal function of the gradient of the solution in L^p of the boundary. We also obtain results with data from Hardy spaces when p=1.

math.AP↗
arXiv2012

The mixed problem for the Lamé system in two dimensions

We consider the mixed problem for $L$ the Lamé system of elasticity in a bounded Lipschitz domain $ Ω\subset\reals ^2$. We suppose that the boundary is written as the union of two disjoint sets, $\partialΩ=D\cup N$. We take traction data from the space $L^p(N)$ and Dirichlet data from a Sobolev space $ W^{1,p}(D)$ and look for a solution $u$ of $Lu =0$ with the given boundary conditions. We give a scale invariant condition on $D$ and find an exponent $ p_0 >1$ so that for $1<p<p_0$, we have a unique solution of this boundary value problem with the non-tangential maximal function of the gradient of the solution in $L^ p(\partialΩ)$. We also establish the existence of a unique solution when the data is taken from Hardy spaces and Hardy-Sobolev spaces with $ p$ in $(p_1,1]$ for some $p_1 <1$.

math.AP↗
arXiv2011

The mixed problem in Lipschitz domains with general decompositions of the boundary

This paper continues the study of the mixed problem for the Laplacian. We consider a bounded Lipschitz domain $Ω\subset \reals^n$, $n\geq2$, with boundary that is decomposed as $\partialΩ=D\cup N$, $D$ and $N$ disjoint. We let $Λ$ denote the boundary of $D$ (relative to $\partialΩ$) and impose conditions on the dimension and shape of $Λ$ and the sets $N$ and $D$. Under these geometric criteria, we show that there exists $p_0>1$ depending on the domain $Ω$ such that for $p$ in the interval $(1,p_0)$, the mixed problem with Neumann data in the space $L^p(N)$ and Dirichlet data in the Sobolev space $W^ {1,p}(D) $ has a unique solution with the non-tangential maximal function of the gradient of the solution in $L^p(\partialΩ)$. We also obtain results for $p=1$ when the Dirichlet and Neumann data comes from Hardy spaces, and a result when the boundary data comes from weighted Sobolev spaces.

math.AP↗