Related papers: Absolute continuity of the best Sobolev constant o…

Let $\Omega$ be a bounded domain of $\mathbf{R}^{N},$ $N\geq2.$ Let, for $p>N,$ \[ \Lambda_{p}(\Omega):=\inf\left\{ \left\Vert \nabla u\right\Vert _{p}^{p}:u\in W_{0}^{1,p}(\Omega)\quad and\quad\left\Vert u\right\Vert _{\infty}=1\right\} .…

Let $\Omega$ be a bounded, smooth domain of $\mathbb{R}^{N},$ $N\geq2.$ For $p>N$ and $1\leq q(p)<\infty$ set \[ \lambda_{p,q(p)}:=\inf\left\{ \int_{\Omega}\left\vert \nabla u\right\vert ^{p}\mathrm{d}x:u\in W_{0}^{1,p}(\Omega)\text{ \ and…

Let $\Omega$ be a bounded, smooth domain of $\mathbb{R}^{N},$ $N\geq2.$ For $1<p<N$ and $0<q(p)<p^{\ast}:=\frac{Np}{N-p}$ let \[ \lambda_{p,q(p)}:=\inf\left\{ \int_{\Omega}\left\vert \nabla u\right\vert ^{p}\mathrm{d}x:u\in…

We study the boundary value problem $-{\rm div}((|\nabla u|^{p_1(x)-2}+|\nabla u|^{p_2(x)-2})\nabla u)=\lambda|u|^{q(x)-2}u$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a bounded domain in $\RR^N$ with smooth boundary,…

We study a minimizing problem associated with the singular problem \[ \left\{ \begin{array} [c]{ll} -\operatorname{div}\left( \left\vert \nabla u\right\vert ^{p-2}\nabla u\right) =\lambda u^{-1} & \mathrm{in\ }\Omega\\ u>0 & \mathrm{in\…

Let $\Omega$ be a bounded, smooth domain of $\mathbb{R}^{N},$ $N\geq1.$ For each $p>N$ we study the optimal function $s=s_{p}$ in the pointwise inequality \[ \left\vert v(x)\right\vert \leq s(x)\left\Vert \nabla v\right\Vert…

Let $\Omega$ be a bounded and smooth domain of $\mathbb{R}^{N}$, $N\geq2$, and consider the eigenvalue problem: $-\Delta_{p}u=\lambda\left| u\right| _{L^{q}(\Omega)}^{p-q}\left| u\right| ^{q-2}u$ in $\Omega,$ $u=0$ on $\partial\Omega,$…

The usual Sobolev inequality in $\mathbb{R}^N$, asserts that $\|\nabla u\|_{L^p(\mathbb{R}^N)} \geq \mathcal{S}\|u\|_{L^{p^*}(\mathbb{R}^N)}$ for $1<p<N$ and $p^*=\frac{pN}{N-p}$, with $\mathcal{S}$ being the sharp constant. Based on a…

This paper is concerned with an explicit value of the embedding constant from $W^{1,q}(\Omega)$ to $L^{p}(\Omega)$ for a bounded domain $\Omega\subset\mathbb{R}^N~(N\in\mathbb{N})$, where $1\leq q\leq p\leq \infty$. To obtain this value, we…

We address the question of attainability of the best constant in the following Hardy-Sobolev inequality on a smooth domain $\Omega$ of \mathbb{R}^n: $$ \mu_s (\Omega) := \inf \{\int_{\Omega}| \nabla u|^2 dx; u \in {H_{1,0}^2(\Omega)}…

We show that independent and uniformly distributed sampling points are as good as optimal sampling points for the approximation of functions from the Sobolev space $W_p^s(\Omega)$ on bounded convex domains $\Omega\subset \mathbb{R}^d$ in…

Let $\Omega \subset \mathbb{R}^d$ be bounded open and connected. Suppose that $W^{1,2}(\Omega) \subset L^r(\Omega)$ for some $r > 2$. Let $A$ be a pure second-order elliptic differential operator with bounded real measurable coefficients on…

Let $\Omega\subset\mathbb{R}^\nu$, $\nu\ge 2$, be a $C^{1,1}$ domain whose boundary $\partial\Omega$ is either compact or behaves suitably at infinity. For $p\in(1,\infty)$ and $\alpha>0$, define \[…

We study existence and regularity properties of solutions to the singular $p$-Laplacean parabolic system in a bounded domain $\Omega$. The main purpose is to prove global $L^r(\varepsilon,T;L^q(\Omega))$, $\varepsilon\geq0$, integrability…

We establish fine bounds for best constants of the fractional subcritical Sobolev embeddings \begin{align*} W_{0}^{s,p}\left(\Omega\right)\hookrightarrow L^{q}\left(\Omega\right), \end{align*} where $N\geq1$, $0<s<1$, $p=1,2$, $1\leq…

We prove the continuity of Sobolev functions $\varphi \in W^{1,n}_{\mathrm{loc}}(\Omega)$, $\Omega \subset \mathbb{R}^n$, that satisfy \[ \lvert\nabla \varphi(x)\rvert^n \le K(x)\bigl(\langle \nabla \varphi(x), \xi(x)\rangle + A(x)\bigr),…

Let $\Omega$ be a smooth, bounded domain of $\mathbb{R}^{N}$, $\omega$ be a positive, $L^{1}$-normalized function, and $0<s<1<p.$ We study the asymptotic behavior, as $p\rightarrow\infty,$ of the pair $\left( \sqrt[p]{\Lambda_{p}%…

Consider a regular domain $\Omega \subset \mathbb{R}^N$ and let $d(x)=\operatorname{dist}(x,\partial\Omega)$. Denote $L^{1,\infty}_a(\Omega)$ the space of functions from $L^{1,\infty}(\Omega)$ having absolutely continuous quasinorms. This…

Let $\Omega $ be an open subset of $\mathbb{R}^{N}$, and let $p,\, q:\Omega \rightarrow \left[ 1,\infty \right] $ be measurable functions. We give a necessary and sufficient condition for the embedding of the variable exponent space…

We consider the eigenvalue problem for the fractional $p \& q-$Laplacian \begin{equation} \left\{\begin{aligned} (- \Delta)_p^{s}\, u + \mu(- \Delta)_q^{s}\, u+ |u|^{p-2}u+\mu|u|^{q-2}u=\lambda\ V(x)|u|^{p-2}u\quad & \text{in } \Omega\\…