Related papers: Approximation of divergence-free vector fields van…

We consider the following basic problem: given an $n$-variate degree-$d$ homogeneous polynomial $f$ with real coefficients, compute a unit vector $x \in \mathbb{R}^n$ that maximizes $|f(x)|$. Besides its fundamental nature, this problem…

We study the boundary weighted regularity of weak solutions $u$ to a $s$-fractional $p$-Laplacian equation in a bounded smooth domain $\Omega$ with bounded reaction and nonlocal Dirichlet type boundary condition, in the singular case…

We consider various notions of vanishing mean oscillation on a (possibly unbounded) domain $\Omega \subset \mathbb{R}^n$, and prove an analogue of Sarason's theorem, giving sufficient conditions for the density of bounded Lipschitz…

In this paper we construct families of bounded domains $\Omega_\varepsilon$ and solutions $u_\varepsilon$ of \[\begin{cases} -\Delta u_\varepsilon=1&\text{ in }\ \Omega_\varepsilon\\ u_\varepsilon=0&\text{ on }\ \partial\Omega_\varepsilon…

We prove that for a bounded simply connected domain $\Omega\subset \mathbb R^2$, the Sobolev space $W^{1,\,\infty}(\Omega)$ is dense in $W^{1,\,p}(\Omega)$ for any $1\le p<\infty$. Moreover, we show that if $\Omega$ is Jordan, then…

This paper proves that the approximation of pointwise derivatives of order $s$ of functions in Sobolev space $W_2^m(\R^d)$ by linear combinations of function values cannot have a convergence rate better than $m-s-d/2$, no matter how many…

The notion of a quasi-free Hilbert module over a function algebra $\mathcal{A}$ consisting of holomorphic functions on a bounded domain $\Omega$ in complex $m$ space is introduced. It is shown that quasi-free Hilbert modules correspond to…

Let $\Omega\subset\mathbb{R}^n$, $n\geq 2$, be a bounded, open and convex set and let $f$ be a positive and non-increasing function depending only on the distance from the boundary of $\Omega$. We consider the $p-$torsional rigidity…

We investigate two density questions for Sobolev, Besov and Triebel--Lizorkin spaces on rough sets. Our main results, stated in the simplest Sobolev space setting, are that: (i) for an open set $\Omega\subset\mathbb R^n$,…

We focus on the Sobolev spaces of bounded subanalytic submanifolds of $\mathbb{R}^n$. We prove that if $M$ is such a manifold then the space $\mathscr{C}_0^\infty(M)$ is dense in $W^{1,p}(M,\partial M)$ (the kernel of the trace operator)…

We consider the minimization problem of $\phi$-divergences between a given probability measure $P$ and subsets $\Omega$ of the vector space $\mathcal{M}_\mathcal{F}$ of all signed finite measures which integrate a given class $\mathcal{F}$…

Let $\Omega\subseteq \mathbb{R}^{d}$ be open and $A$ a complex uniformly strictly accretive $d\times d$ matrix-valued function on $\Omega$ with $L^{\infty}$ coefficients. Consider the divergence-form operator ${\mathscr L}^{A}=-{\rm…

Concerned with the Stokes systems with rapidly oscillating periodic coefficients, we mainly extend the recent works in \cite{SGZWS,G} to those in term of Lipschitz domains. The arguments employed here are quite different from theirs, and…

This paper deals with the fractional Sobolev spaces $W^{s, p}(\Omega)$, with $s\in (0, 1]$ and $p\in[1,+\infty]$. Here, we use the interpolation results in [4] to provide suitable conditions on the exponents $s$ and $p$ so that the spaces…

We approximate functions defined on smooth bounded domains by elements of the eigenspaces of the Laplacian or the Stokes operator in such a way that the approximations are bounded and converge in both Sobolev and Lebesgue spaces. We prove…

Let $\Omega\subset\mathbb{C}^n$ be a strictly pseudoconvex Runge domain with $C^2$-smooth defining function, $l\in\mathbb{N},$ $p\in(1,\infty).$ We prove that the holomorphic function $f$ has derivatives of order $l$ in $H^p(\Omega)$ if and…

This paper establishes a bivariate Hardy-Sobolev inequality. Let $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) be an open domain, $s \in (0,2)$, $\alpha > 1$, $\beta > 1$ with $\alpha + \beta = 2^*(s)$, and $\kappa \in \mathbb{R}$. For any…

We establish the first extension results for divergence-free (or solenoidal) elements of $\mathrm{L}^{1}$-based function spaces. Here, the key point is to preserve the solenoidality constraint while simultaneously keeping the underlying…

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain and $1 < p < \infty$. We characterize $(1,p)$-extension domains in terms of inequalities of Bourgain--Brezis--Mironescu type. More precisely, we show that $\Omega$ is a $(1,p)$-extension…

We investigate the form of the closure of the smooth, compactly supported functions $C_{c}^{\infty}(\Omega)$ in the weighted fractional Sobolev space $W^{s,p;\,w,v}(\Omega)$ for bounded $\Omega$. We focus on the weights $w,\,v$ being powers…