Related papers: Sharp quantitative stability of the planar Brunn-M…

We consider the inhomogeneous biharmonic nonlinear Schr\"odinger equation $$ i u_t +\Delta^2 u+\lambda|x|^{-b}|u|^\alpha u = 0, $$ where $\lambda=\pm 1$ and $\alpha$, $b>0$. In the subctritical case, we improve the global well-posedness…

In this paper we address the global stability problem for double-bubbles in the plane. This is accomplished by combining the "improved convergence theorem" for planar clusters developed in arXiv:1409.6652 with an ad hoc analysis of the…

We study the quantitative stability for the classical Brezis-Nirenberg problem associated with the critical Sobolev embedding $H^1_0(\Omega) \hookrightarrow L^{\frac{2n}{n-2}}(\Omega)$ in a smooth bounded domain $\Omega \subset…

We study the dimensional Brunn-Minkowski inequality for even log-concave probability measures $\mu$ on $\mathbb{R}^n$ via an analytic approach based on diffusion operators and gradient estimates. Our main result asserts that for every pair…

In the case of symmetries with respect to n independent linear hyperplanes, the stability of the solution of the Logarithmic Minkowski problem on S^{n-1} is established.

By developing a unified approach based on integral representations, we establish sharp quantitative stability estimates for critical points of the fractional Sobolev inequalities induced by the embedding $\dot{H}^s({\mathbb R}^n)…

In this paper, we establish several improved Caffarelli-Kohn-Nirenberg and Hardy-type inequalities. Our main results are divided into two parts. In the first part, we consider the following Caffarelli-Kohn-Nirenberg inequality:…

For a Minkowski centered convex compact set $K$ we define $\alpha(K)$ to be the smallest possible factor to cover $K \cap (-K)$ by a rescalation of $\mathrm{conv} (K\cup (-K))$ and give a complete description of the possible values of…

We prove a strong form of the quantitative Sobolev inequality in $\mathbb{R}^n$ for $p\geq 2$, where the deficit of a function $u\in \dot W^{1,p} $ controls $\| \nabla u -\nabla v\|_{L^p}$ for an extremal function $v$ in the Sobolev…

In this work, the $L_p$ version (for $p> 1$) of the dimensional Brunn-Minkowski inequality for the standard Gaussian measure $\gamma_n(\cdot)$ on $\mathbb{R}^n$ is shown. More precisely, we prove that for any $0$-symmetric convex sets with…

Let $m(G)$ be the infimum of the volumes of all open subgroups of a unimodular locally compact group $G$. Suppose integrable functions $\phi_1 , \phi_2 \colon G \to [0,1]$ satisfy $\| \phi_1 \| \leq \| \phi_2 \|$ and $\| \phi_1 \| + \|…

We prove that the log-Brunn-Minkowski inequality \begin{equation*} |\lambda K+_0 (1-\lambda)L|\geq |K|^{\lambda}|L|^{1-\lambda} \end{equation*} (where $|\cdot|$ is the Lebesgue measure and $+_0$ is the so-called log-addition) holds when…

In this paper, we consider the following variational problem: \begin{eqnarray*} \inf_{u\in…

We prove the stability of the Einstein-scalar field Lichnerowicz equation under subcritical perturbations of the critical nonlinearity in dimensions 3, 4 and 5. As a consequence, we obtain the existence of a second solution to the equation…

$L_p$ Brunn-Minkowski type inequa\-li\-ties for the lattice point enumerator $\mathrm{G}_n(\cdot)$ are shown, both in a geometrical and in a functional setting. In particular, we prove that \[\mathrm{G}_n\bigl((1-\lambda)\cdot K +_p…

The goal of this short note is to prove qualitative stability for a family of trace Sobolev inequalities first proven by Carlen \& Loss for $p=2$ and by Maggi and the author for $p\in (1,n)$. This answers an open problem raised in a recent…

Extending results of Linial (1984) and Aigner (1985), we prove a uniform lower bound on the balance constant of a poset $P$ of width $2$. This constant is defined as $\delta(P) = \max_{(x, y)\in P^2}\min\{\mathbb{P}(x\prec y),…

We describe the harmonic interpolation of convex bodies, and prove a strong form of the Brunn-Minkowski inequality and characterize its equality case. As an application we improve a theorem of Berndtsson on the volume of slices of a…

We prove definitive results on the global stability of the flat space among solutions of the Einstein-Klein-Gordon system. Our main theorems in this monograph include: (1) A proof of global regularity (in wave coordinates) of solutions of…

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…