Related papers: On directional Whitney inequality

Wang and Ye conjectured in [22]: Let $\Omega$ be a regular, bounded and convex domain in $\mathbb{R}^{2}$. There exists a finite constant $C({\Omega})>0$ such that \[ \int_{\Omega}e^{\frac{4\pi u^{2}}{H_{d}(u)}}dxdy\le C(\Omega),\;\;\forall…

Let $\Omega $ be a bounded ${\mathcal{C}}^{\infty}$-smoothly bounded domain in ${\mathbb{C}}^{n}.$ For such a domain we define a new notion between strict pseudo-convexity and pseudo-convexity: the size of the set $W$ of weakly…

We give two results about Harnack type inequalities. First, on compact smooth Riemannian surface without boundary, we have an estimate of the type $\sup +\inf$. The second result concerns the solutions of prescribed scalar curvature…

The paper present new definitions for weighted DT moduli. Similarly, we a general outcome in an equivalence of moduli of smoothness are obtained. It is known that, any $r \in \mathbb{N}_{\circ}$ , $0<p \leq \infty$, $1 \leq \eta \leq r$ and…

We construct a sequence $\{\Sigma_\ell\}_{\ell=1}^\infty$ of closed, axially symmetric surfaces $\Sigma_\ell\subset \mathbb{R}^3$ that converges to the unit sphere in $W^{2,p}\cap C^1$ for every $p\in[1,\infty)$ and such that, for every…

For $\Omega \subset \mathbb{R}^n$, a convex and bounded domain, we study the spectrum of $-\Delta_\Omega$ the Dirichlet Laplacian on $\Omega$. For $\Lambda\geq0$ and $\gamma \geq 0$ let $\Omega_{\Lambda, \gamma}(\mathcal{A})$ denote any…

Let $\Omega \subset \mathbb{R}^n$ be a convex domain and let $f:\Omega \rightarrow \mathbb{R}$ be a positive, subharmonic function (i.e. $\Delta f \geq 0$). Then $$ \frac{1}{|\Omega|} \int_{\Omega}{f dx} \leq \frac{c_n}{ |\partial \Omega| }…

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…

Finite difference methods (FDMs) are widely used for solving partial differential equations (PDEs) due to their relatively simple implementation. However, they face significant challenges when applied to non-rectangular domains and in…

This paper deals with the lack of compactness in nonlinear elliptic problems $(P)$. In particular, a domain $\Omega$ is provided where not converging Palais-Smale sequences exist at every energy level. Nevertheless, it is proved that…

We prove uniqueness in the inverse conductivity problem for uniformly elliptic conductivities in $W^{s,p}(\Omega)$, where $\Omega \subset \mathbb R^n$ is Lipschitz, $3\leq n \leq 6$, and $s$ and $p$ are such that $ W^{s,p}(\Omega)\not…

We consider shape functionals of the form $F_q(\Omega)=P(\Omega)T^q(\Omega)$ on the class of open sets of prescribed Lebesgue measure. Here $q>0$ is fixed, $P(\Omega)$ denotes the perimeter of $\Omega$ and $T(\Omega)$ is the torsional…

In this paper, we mainly study error bounds for a single convex inequality and semi-infinite convex constraint systems, and give characterizations of stability of error bounds via directional derivatives. For a single convex inequality, it…

A Hardy inequality of the form \[\int_{\tilde{\Omega}} |\nabla f({\bf{x}})|^p d {\bf{x}} \ge (\frac{p-1}{p})^p \int_{\tilde{\Omega}} \{1 + a(\delta, \partial \tilde{\Omega})(\x)\}\frac{|f({\bf{x}})|^p}{\delta({\bf{x}})^p} d{\bf{x}}, \] for…

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…

In this paper we are interested in possible extensions of an inequality due to Minkowski: $\int_{\partial\Omega} H\,dA \geq \sqrt{4\pi A(\partial\Omega)}$ valid for any regular open set $\Omega\subset\mathbb{R}^3$, where $H$ denotes the…

A smooth rigidity inequalitiy provides an explicit lower bound for the $(d+1)$-st derivatives of a smooth function $f$, which holds, if $f$ exhibits certain patterns, forbidden for polynomials of degree $d$. The main goal of the present…

Aim of this paper is to prove necessary and sufficient conditions on the geometry of a domain $\Omega \subset \mathbb{R}^n$ in order that the homogeneous Dirichlet problem for the infinity-Laplace equation in $\Omega$ with constant source…

We prove an improvement for the sharp Adams inequality in $W^{m,\frac nm}_0(\Omega)$ where $\Omega$ is a bounded domain in $\mathbb{R}^n$ inspired by Lions Concentration--Compactness principle for the sharp Moser--Trudinger inequality. Our…

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…