相关论文: A sharp Trudinger-Moser inequality on any bounded …

The Trudinger-Moser inequality states that for functions $u \in H_0^{1,n}(\Omega)$ ($\Omega \subset \mathbb R^n$ a bounded domain) with $\int_\Omega |\nabla u|^ndx \le 1$ one has $\int_\Omega (e^{\alpha_n|u|^{\frac n{n-1}}}-1)dx \le c…

Let $\Omega$ be a smooth bounded domain in $\mathbf R^2$ and $\lambda^{\mathsf N} (\Omega)$ the first non-zero Neumann eigenvalue of the operator $-\Delta$ on $\Omega$. In this paper, for any $\gamma \in [0, \lambda^{\mathsf N} (\Omega) )$,…

A property of smooth convex domains $\Omega \subset \mathbb{R}^n$ is that if two points on the boundary $x, y \in \partial \Omega$ are close to each other, then their normal vectors $n(x), n(y)$ point roughly in the same direction and this…

Let $\Omega\subset\mathbb{R}^2$ be a smooth bounded domain with $0\in\partial\Omega$. In this paper, we prove that for any $\beta\in(0,1)$, the supremum $$\sup_{u\in W^{1,2}(\Omega), \int_\Omega u dx=0, \int_\Omega|\nabla…

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)}…

The Hardy constant of a simply connected domain $\Omega\subset\mathbf{R}^2$ is the best constant for the inequality \[ \int_{\Omega}|\nabla u|^2dx \geq c\int_{\Omega} \frac{u^2}{{\rm dist}(x,\partial\Omega)^2}\, dx \; , \;\;\quad u\in…

In this paper, on a compact Riemann surface $(\Sigma, g)$ with smooth boundary $\partial\Sigma$, we concern a Trudinger-Moser inequality with mean value zero. To be exact, let $\lambda_1(\Sigma)$ denotes the first eigenvalue of the…

In this paper we study the following torsion problem \begin{equation*} \begin{cases} -\Delta u=1~&\mbox{in}\ \Omega,\\[1mm] u=0~&\mbox{on}\ \partial\Omega. \end{cases} \end{equation*} Let $\Omega\subset \mathbb{R}^2$ be a bounded, convex…

The Hardy constant of a simply connected domain $\Omega\subset\R^2$ is the best constant for the inequality \[ \int_{\Omega}|\nabla u|^2dx \geq c\int_{\Omega} \frac{u^2}{{\rm dist}(x,\partial\Omega)^2}\, dx \;, u\in C^{\infty}_c(\Omega). \]…

We consider H\"older continuous weak solutions $u\in C^\gamma(\Omega)$, $u\cdot n|_{\partial \Omega}=0$, of the incompressible Euler equations on a bounded and simply connected domain $\Omega\subset\mathbb{R}^d$. If $\Omega$ is of class…

Let $\Omega$ be an open, bounded domain in the plane with connected and smooth boundary, and $\omega$ an eigenfunction of the Neumann Laplacian corresponding to some Neumann eigenvalue $\mu > 0$. If the boundary value of $\omega$ is a…

Alexandrov's estimate states that if $\Omega$ is a bounded open convex domain in ${\mathbb R}^n$ and $u:\bar \Omega\to {\mathbb R}$ is a convex solution of the Monge-Ampere equation $\det D^2 u = f$ that vanishes on $\partial \Omega$, then…

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 $\Omega\subset{\mathbb R}^2$ be a bounded domain on which Hardy's inequality holds. We prove that $[\exp(u^2)-1]/\delta^2\in L^1(\Omega)$ if $u\in H^1_0(\Omega)$, where $\delta$ denotes the distance to $\partial\Omega$. The…

The goal of this note is to show that, also in a bounded domain $\Omega \subset \mathbb{R}^n$, with $\partial \Omega\in C^2$, any weak solution, $(u(x,t),p(x,t))$, of the Euler equations of ideal incompressible fluid in $\Omega\times (0,T)…

The classical Poincar\'e inequality establishes that for any bounded regular domain $\Omega\subset \R^N$ there exists a constant $C=C(\Omega)>0$ such that $$ \int_{\Omega} |u|^2\, dx \leq C \int_{\Omega} |\nabla u|^2\, dx \ \ \forall u \in…

Continuing our previous work (Cohn, Lam, Lu, Yang, Nonlinear Analysis (2011), doi: 10.1016 /j.na.2011.09.053), we obtain a class of Trudinger-Moser inequalities on the entire Heisenberg group, which indicate what the best constants are. All…

For any bounded convex domain $\Omega$ with $C^{2}$ boundary in $\mathbb{C}^{n}$, we show that there exist positive constants $C_{1}$ and $C_{2}$ such that \[ C_{1}\sqrt{\dfrac{K\left(w,w\right)}{\delta\left(w\right)}}\leq\left\Vert…

Refining the sharp upper bounds $\mu_{k,d}^* $ obtained by Kr\"oger (1999) for the $k$-th Neumann eigenvalue of a convex domain $\Omega \subset \mathbb{R}^d$, we prove the following inequalities: for any $k\in \mathbb{N}$ there exists a…

Let $\Omega$ be a bounded, convex, centrally symmetric in $\mathbb{R}^{2}$ with a connected $C^{2,\epsilon}$ ($\epsilon\in(0,1)$) boundary. We show that, if the following overdetermined elliptic problem \begin{equation} -\Delta u=\alpha…