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

We study the following version of Hardy-type inequality on a domain $\Omega$ in a Riemannian manifold $(M,g)$: $$ \int{\Omega}|\nabla u|_g^p\rho^\alpha dV_g \geq \left(\frac{|p-1+\beta|}{p}\right)^p\int{\Omega}\frac{|u|^p|\nabla…

Let $\Omega\subset\mathbb R^n$ be a Lipschitz domain. We prove that, $\Omega$ satisfies the following Serrin-type overdetermined system $$u \in W^{1,2}(\mathbb R^n), \quad u=0\ \text{ a.e. in }\mathbb R^n\setminus \Omega,\quad \Delta…

For $N\geq 5$ and $0<\mu<N-4$, we first show a non-degenerate result of the extremal functions for the following Rellich-Sobolev type inequality \begin{align*} \int_{\mathbb{R}^N}|\Delta u|^2 \mathrm{d}x…

We will show in this paper that if $\lambda$ is very close to 1, then $$I(M,\lambda,m)= \sup_{u\in H^{1,n}_0(M) ,\int_M|\nabla u|^ndV=1}\int_\Omega (e^{\alpha_n |u|^\frac{n}{n-1}}-\lambda\sum\limits_{k=1}^m\frac{|\alpha_nu^\frac{n}{n-1}|^k}…

We prove that if $\Omega\subset \mathbb R^n$ is a bounded open set and $n\alpha> {\rm dim}_b (\partial \Omega) = d$, then the Brouwer degree deg$(v,\Omega,\cdot)$ of any H\"older function $v\in C^{0,\alpha}\left (\Omega, \mathbb…

We study the interior regularity of solutions to the Dirichlet problem $Lu=g$ in $\Omega$, $u=0$ in $\R^n\setminus\Omega$, for anisotropic operators of fractional type $$ Lu(x)= \int_{0}^{+\infty}\,d\rho \int_{S^{n-1}}\,da(\omega)\, \frac{…

We consider the imbedding inequality || f ||_{L^r(R^d)} <= S_{r,n,d} || f ||_{H^{n}(R^d)}; H^{n}(R^d) is the Sobolev space (or Bessel potential space) of L^2 type and (integer or fractional) order n. We write down upper bounds for the…

In this paper we obtain an inequality on the unit disc $B$ in the plane, which improves the classical Moser-Trudinger inequality and the classical Hardy inequality at the same time. Namely, there exists a constant $C_0>0$ such that \[…

We study the complete K\"{a}hler-Einstein metric of a Hartogs domain $\widetilde {\Omega}$, which is obtained by inflation of an irreducible bounded symmetric domain $\Omega $, using a power $N^{\mu}$ of the generic norm of $\Omega$. The…

We study the Dirichlet problem $-\div(|\nabla u|^{p(x)-2} \nabla u) =0 $ in $\Omega$, with $u=f$ on $\partial \Omega$ and $p(x) = \infty$ in $D$, a subdomain of the reference domain $\Omega$. The main issue is to give a proper sense to what…

We consider Laplacian eigenfunctions on a domain $\Omega \subset \mathbb{R}^d$. Under Neumann boundary conditions, the first eigenfunction is constant and the others have mean value 0. The situation is different for Dirichlet boundary…

We construct a bounded $C^{1}$ domain $\Omega$ in $R^{n}$ for which the $H^{3/2}$ regularity for the Dirichlet and Neumann problems for the Laplacian cannot be improved, that is, there exists $f$ in $C^{\infty}(\overline\Omega)$ such that…

We derive a decomposition result for regular, two-dimensional domains into John domains with uniform constants. We prove that for every simply connected domain $\Omega \subset {\Bbb R}^2$ with $C^1$-boundary there is a corresponding…

We consider the Dirichlet problem for stationary biharmonic maps $u$ from a bounded, smooth domain $\Omega\subset\mathbb R^n$ ($n\ge 5$) to a compact, smooth Riemannian manifold $N\subset\mathbb R^l$ without boundary. For any smooth…

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\…

Consider the following inequalities due to Caffarelli, Kohn and Nirenberg {\it (Compositio Mathematica,1984):} $$\Big(\int_\Omega \frac{|u|^r}{|x|^s}dx\Big)^{\frac{1}{r}}\leq C(p,q,r,\mu,\sigma,s)\Big(\int_\Omega \frac{|\nabla…

We study the problem of finding a function u verifying --$\Delta$u = 0 in $\Omega$ under the boundary condition $\partial$u $\partial$n + g(u) = $\mu$ on $\partial$$\Omega$ where $\Omega$ $\subset$ R N is a smooth domain, n the normal unit…

Let $\mathbb{S} \subset \mathbb{C}$ be the circle in the plane, and let $\Omega: \mathbb{S} \to \mathbb{S}$ be an odd bi-Lipschitz map with constant $1+\delta_\Omega$, where $\delta_\Omega>0$ is small. Assume also that $\Omega$ is twice…

In this paper, we prove the existence of nontrivial unbounded domains $\Omega\subset\mathbb{R}^{n+1},n\geq1$, bifurcating from the straight cylinder $B\times\mathbb{R}$ (where $B$ is the unit ball of $\mathbb{R}^n$), such that the…

In this note two results are established for energy functionals that are given by the integral of $ W(\mathbf x,\nabla \mathbf u(\mathbf x))$ over $\Omega \subset\mathbb{R}^n$ with $\nabla \mathbf u \in BMO(\Omega;{\mathbb R}^{N\times n})$,…