Related papers: A sharp Trudinger-Moser type inequality for unboun…

Let $\Omega \subset \mathbb{R}^d$ be a bounded domain and let $\lambda_1, \lambda_2, \dots$ denote the sequence of eigenvalues of the Laplacian subject to Dirichlet boundary conditions. We consider inequalities for $\lambda_n$ that are…

Let $n\ge 2$ and $s\in (n-2,n)$. Assume that $\Omega\subset \mathbb{R}^n$ is a one-sided bounded non-tangentially accessible domain with $s$-Ahlfors regular boundary and $\sigma$ is the surface measure on the boundary of $\Omega$, denoted…

We establich quantitative stability estimates for the Trudinger-Moser inequality on smooth, bounded domains in the Euclidean plane. More specifically, we prove that the deficit in the Trudinger-Moser inequality quadratically controls the…

In this paper we prove a Sobolev and a Morrey type inequality involving the mean curvature and the tangential gradient with respect to the level sets of the function that appears in the inequalities. Then, as an application, we establish…

We establish embeddings on a class of Sobolev spaces with potential weights on unbounded domains. Our results provide embeddings into weighted Lebesgue spaces $L^q_\theta$ with radial power weights and establish the existence and…

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…

Let $n \geq 2$ and $\Omega \subset \mathbb{R}^n$ be a bounded domain. Then by Trudinger-Moser embedding, $W_0^{1,n}(\Omega)$ is embedded in an Orlicz space consisting of exponential functions. Consider the corresponding semi linear…

Let $F$ be convex and homogeneous of degree $1$, its polar $F^{o}$ represent a finsler metric on $\mathbb{R}^{n}$, and $\Omega$ be any bounded open set in $\mathbb{R}^{n}$. In this paper, we first construct the theoretical structure of…

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…

Let $\Omega$ be an open set in Euclidean space $\R^m,\, m=2,3,...$, and let $v_{\Omega}$ denote the torsion function for $\Omega$. It is known that $v_{\Omega}$ is bounded if and only if the bottom of the spectrum of the Dirichlet Laplacian…

We consider the quasilinear parabolic-parabolic Keller-Segel system $$ u_t=\nabla \cdot (D(u)\nabla u) - \nabla \cdot (S(u)\nabla v), \qquad x\in\Omega, \ t>0, v_t=\Delta v -v + u, x\in\Omega, \ t>0, $$ under homogeneous Neumann boundary…

Given a sufficiently symmetric domain $\Omega\Subset\mathbb{R}^2$, for any $k\in \mathbb{N}\setminus \{0\}$ and $\beta>4\pi k$ we construct blowing-up solutions $(u_\varepsilon)\subset H^1_0(\Omega)$ to the Moser-Trudinger equation such…

It has been well known that if $\Omega$ is a bounded $C^1$-domain in $\R^n,\ n \ge 2$, then for every Radon measure $f$ on $\Omega$ with finite total variation, there exists a unique weak solution $u\in W_0^{1,1}(\Omega )$ of the Poisson…

Let {\Omega} be a bounded domain in R^n with C^{1,1} boundary and let u_{\lambda} be a Neumann Laplace eigenfunction in {\Omega} with eigenvalue {\lambda}. We show that the (n - 1)-dimensional Hausdorff measure of the zero set of…

Let $\Omega \subset \mathbb R^d$ be a $C^1$ domain or, more generally, a Lipschitz domain with small Lipschitz constant and $A(x)$ be a $d \times d$ uniformly elliptic, symmetric matrix with Lipschitz coefficients. Assume $u$ is harmonic in…

We consider the 2D incompressible Euler equation on a bounded simply connected domain $\Omega$. We give sufficient conditions on the domain $\Omega$ so that for all initial vorticity $\omega_0 \in L^{\infty}(\Omega)$ the weak solutions are…

Let $ m, n $ be integers such that $ \frac{n}{2} > m \geq 1 $ and let $ (M, g) $ be a closed $ n-$dimensional Riemannian manifold. We prove there exists some $ B \in \mathbb{R} $ depending only on $ (M, g) $, $ m $, and $ n $ such that for…

The main purpose of this paper is to prove a sharp Sobolev inequality in an exterior of a convex bounded domain. There are two ingredients in the proof: One is the observation of some new isoperimetric inequalities with partial free…

Let $\mu_2(\Omega)$ be the first positive eigenvalue of the Neumann Laplacian in a bounded domain $\Omega\subset\mathbb{R}^N$. It was proved by Szeg\H{o} for $N=2$ and by Weinberger for $N \geq 2$ that among all equimeasurable domains…

We establish -among other things- existence and multiplicity of solutions for the Dirichlet problem $\sum_i\partial_{ii}u+\frac{|u|^{\crit-2}u}{|x|^s}=0$ on smooth bounded domains $\Omega$ of $ \rn$ ($n\geq 3$) involving the critical…