Related papers: Characterization of functions with zero traces via…

We discuss the solvability of Dirichlet problems of the type $- \Delta_{p, w} u = \sigma$ in $\Omega$; $u = 0$ on $\partial \Omega$, where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$, $\Delta_{p, w}$ is a weighted $(p, w)$-Laplacian…

Given a connected Lipschitz domain U we let L(U) be the subset of functions in 2nd order Sobolev space whose gradient (in the sense of trace) is equal to the inward pointing unit normal to U. The the Aviles Giga functional over L(U) serves…

The $L^p$-spaces, with $p \not = \infty$, form a partial algebra $(L^p(\Omega), \Gamma, \cdot)$ with pointwise multiplication of functions. The Sobolev spaces $W^{k,p}(\Omega)$, delineated by weak derivatives as subspaces of $L^p$-spaces is…

In this paper, we investigate the properties of linear operators defined on $L^p(\Omega)$ that are the composition of differential operators with functions that vanish on the boundary $\partial \Omega$. We focus on bounded domains $\Omega…

We introduce a non-linear criterion which allows us to determine when a function can be written as a sum of functions belonging to homogeneous fractional spaces: for $\ell \in \mathbb{N}^*$, $s_i\in (0, 1)$ and $p_i \in [1, +\infty)$, $u :…

A complete description of traces on $\mathbb{R}^{n}$ of functions from the weighted Sobolev space $W^{l}_{1}(\mathbb{R}^{n+1},\gamma)$, $l \in \mathbb{N}$, with weight $\gamma \in A^{\rm loc}_{1}(\mathbb{R}^{n+1})$ is obtained. In the case…

Let $\Omega$ be a subanalytic bounded open subset of $\mathbb{R}^n$, with possibly singular boundary. We show that given $p\in [1,\infty)$, there is a constant $C$ such that for any $u\in W^{1,p}(\Omega)$ we have $||u-u_{\Omega}||_{L^p} \le…

We study positive solutions to the problem $-\Delta_p u + \vartheta |\nabla u|^q = \frac{1}{u^\gamma} + f(u)$ in $\mathbb{R}^N_+$ with the zero Dirichlet boundary condition, where $p>1$, $\gamma>0$, $0<q\le p$, $\vartheta\ge0$ and…

We consider the problem $$ (P_\lambda)\quad -\Delta_{p}u=\lambda u^{p-1}+a(x)u^{q-1},\quad u\geq0\quad\mbox{ in }\Omega $$ under Dirichlet or Neumann boundary conditions. Here $\Omega$ is a smooth bounded domain of $\mathbb{R}^{N}$…

For $m \geq 2$, let $(\mathbb{Z}_{m+1}^N, |\cdot|)$ denote the group equipped with the so-called $l^0$ metric, \[ |y| = \left| \big( y(1), \dots, y(N) \big) \right| := | \{1 \leq i \leq N : y(i) \neq 0 \} |,\] and define the…

Given a bounded domain $\Omega \subset \mathbb{R}^n$, a result by Bourgain, Brezis, and Mironescu characterizes when a function $f \in L^p(\Omega)$ is in the Sobolev space $W^{1,p}(\Omega)$ based on the limiting behavior of its Besov…

For $1<p<\infty$ we give a characterization of the Sobolev space $\dot W^{1,p}(\mathbb R^d)$ in terms of the oscillations of a function on balls of varying centers and radii. Our work is motivated both by the study of trace ideal properties…

Given any uniform domain $\Omega$, the Triebel-Lizorkin space $F^s_{p,q}(\Omega)$ with $0<s<1$ and $1<p,q<\infty$ can be equipped with a norm in terms of first order differences restricted to pairs of points whose distance is comparable to…

Given any divergence-free vector field of Sobolev class $W^{m,p}_0(\Omega)$ in a bounded open subset $\Omega \subset \mathbb{R}^2$, we are interested in approximating it in the $W^{m,p}$ norm with divergence-free smooth vector fields…

We provide relaxation for not lower semicontinuous supremal functionals of the type $W^{1,\infty}(\Omega;\mathbb R^d) \ni u \mapsto\supess_{ x \in \Omega}f(\nabla u(x))$ in the vectorial case, where $\Omega\subset \mathbb R^N$ is a…

Let $\Omega \Subset \mathbb R^n$, $f \in C^1(\mathbb R^{N\times n})$ and $g\in C^1(\mathbb R^N)$, where $N,n \in \mathbb N$. We study the minimisation problem of finding $u \in W^{1,\infty}_0(\Omega;\mathbb R^N)$ that satisfies \[ \big\|…

Let $\Omega$ be a domain in $\mathbb{R}^d$, $d\geq 2$, and $1<p<\infty$. Fix $V\in L_{\mathrm{loc}}^\infty(\Omega)$. Consider the functional $Q$ and its G\^{a}teaux derivative $Q^\prime$ given by $$Q(u):=\int_\Omega (|\nabla…

We establish a pointwise limit theorem for a broad class of pa\-ra\-me\-ter-\-de\-pen\-dent BMO-type seminorms as the parameter tends to zero. By introducing novel BMO-type seminorms, we provide a unified framework that extends several…

Let $\Omega\subset\mathbb R^n$ be a Lipschitz domain. Given $1\leq p<k\leq n$ and any $u\in W^{2,p}(\Omega)$ belonging to the little H\"older class $c^{1,\alpha}$, we construct a sequence $u_j$ in the same space with…

This paper examines the solvability of the equation $\mathrm{div} \ \mathbf{u} = f$ with a zero Dirichlet boundary condition for $\mathbf{u}$. A classical result establishes that for a bounded domain $\Omega \subset \mathbb{R}^N$ with a…