Related papers: Blowing up extremal Poincar\'e type manifolds

We investigate the behaviour of radial solutions to the Lin-Ni-Takagi problem in the ball $B_R \subset \mathbb{R}^N$ for $N \ge 3$: \begin{equation*} \left \{ \begin{aligned} - \triangle u_p + u_p & = |u_p|^{p-2}u_p & \textrm{ in } B_R, \\…

Let D a divisor with simple normal crossings in a Kahler manifold X. The purpose of this short note is to show that the existence of a Poincare type metric with constant scalar curvature in on the complement of D implies for any component…

We consider periodically perforated unbounded open sets and prove existence of extremals for the relevant sharp Poincar\'e-Sobolev embedding constant. The existence result holds no matter the shape or the regularity of the hole: it is…

Let $(M,J)$ be a $2n$-dimensional almost complex manifold and let $x\in M$. We define the notion of almost complex blow-up of $(M,J)$ at $x$. We prove the existence of almost complex blow-ups at $x$ under suitable assumptions on the almost…

In this paper, we prove several Poincar\'e inequalities of fractional type on conformally flat manifolds with finite total Q-curvature. This shows a new aspect of the $Q$-curvature on noncompact complete manifolds.

This paper provides details of the construction, properties and some applications of the ambient metric associated to a conformal class of metrics on a smooth manifold. Existence and uniqueness of formal expansions defining such metrics are…

Intersection homology with coefficients in a field restores Poincar\'e duality for some spaces with singularities, as pseudomanifolds. But, with coefficients in a ring, the behaviours of manifolds and pseudomanifolds are different. This…

We consider the problem v_t & = \Delta v+ |v|^{p-1}v \quad\hbox{in }\ \Omega\times (0, T), v & =0 \quad\hbox{on } \partial \Omega\times (0, T ) , v& >0 \quad\hbox{in }\ \Omega\times (0, T) . In a domain $\Omega\subset \mathbb R^d$, $d\ge 7$…

Perturbations of super Poincar\'e and weak Poincar\'e inequalities for L\'evy type Dirichlet forms are studied. When the range of jumps is finite our results are natural extensions to the corresponding ones derived earlier for diffusion…

In this paper, we show several vanishing type theorems for $p$-harmonic $\ell$-forms on Riemannian manifolds ($p\geq2$). First of all, we consider complete non-compact immersed submanifolds $M^n$ of ${N}^{n+m}$ with flat normal bundle, we…

The John-Nirenberg theorem states that functions of bounded mean oscillation are exponentially integrable. In this article we give two extensions of this theorem. The first one relates the dyadic maximal function to the sharp maximal…

We consider the nonlinear Schr\"odinger equation with $L^2$-critical exponent and an inhomogeneous damping term. By using the tools developed by Merle and Raphael, we prove the existence of blowup phenomena in the energy space…

We prove a sharp Poincar\'e inequality for subsets $\Omega$ of (essentially non-branching) metric measure spaces satisfying the Measure Contraction Property $\textrm{MCP}(K,N)$, whose diameter is bounded above by $D$. This is achieved by…

We introduce a spectrum of monotone coarse invariants for metric measure spaces called Poincar\'{e} profiles. The two extremes of this spectrum determine the growth of the space, and the separation profile as defined by…

In this paper, using special metric deformations introduced by Aubin, we construct Riemannian metrics satisfying non-vanishing conditions concerning the Weyl tensor, on every compact manifold. In particular, in dimension four, we show that…

We introduce high staircase infinite measure preserving transformations and prove that they are mixing under a restricted growth condition. This is used to (i) realize each subset $E\subset\Bbb N\cup\{\infty\}$ as the set of essential…

Let $M$ be a compact K\"ahler manifold and $N$ be a subvariety with codimension greater than or equal to 2. We show that there are no complete K\"ahler--Einstein metrics on $M-N$. As an application, let $E$ be an exceptional divisor of $M$.…

We show that for a given holomorphic noncharacteristic surface S in two-dimensional complex space, and a given holomorphic function on S, there exists a unique meromorphic solution of Burgers' equation which blows up on S. This proves the…

The blow-up for semilinear wave equations with the scale invariant damping has been well-studied for sub-Fujita exponent. However, for super-Fujita exponent, there is only one blow-up result which is obtained in 2014 by Wakasugi in the case…

It is still not known whether a solution to the incompressible Euler equation, endowed with a smooth initial value, can blow-up in finite time. In [{\em Comm. Math. Phys.}, 378:557--568, 2020] it has been shown that, if it exists, such a…