Related papers: Quantitative stability of a nonlocal Sobolev inequ…

We study Sobolev mappings $f \in W_{\mathrm{loc}}^{1,n} (\mathbb{R}^n, \mathbb{R}^n)$, $n \ge 2$, that satisfy the heterogeneous distortion inequality \[\left|Df(x)\right|^n \leq K J_f(x) + \sigma^n(x) \left|f(x)\right|^n\] for almost every…

In this paper, we consider the following Schr\"{o}dinger equation: \begin{equation*} \begin{cases} -\Delta u=\lambda u+\mu|u|^{q-2}u+|u|^{2^*-2}u\quad\text{in }\mathbb{R}^N,\\ \int_{\mathbb{R}^N}|u(x)|^2dx=a,\quad u\in H^1(\mathbb{R}^N),\\…

Let $\Omega$ be a bounded domain of $\mathbb{R}^{N+1}$ ($N \geq 3$) with smooth boundary $\partial \Omega$ and $\Sigma$ be a closed submanifold contained on $\partial \Omega$ and containing $0$. We are interesting in the existence of…

In this paper, we consider the upper critical Choquard equation with a local perturbation \begin{equation*} \begin{cases} -\Delta u=\lambda u+(I_\alpha\ast|u|^{p})|u|^{p-2}u+\mu|u|^{q-2}u,\ x\in \mathbb{R}^{N},\\ u\in H^1(\mathbb{R}^N),\…

We study uniform Sobolev inequalities for the second order differential operators $P(D)$ of non-elliptic type. For $d\ge3$ we prove that the Sobolev type estimate $\|u\|_{L^q(\mathbb{R}^d)}\le C \|P(D)u\|_{L^p(\mathbb{R}^d)}$ holds with $C$…

We establish the optimal asymptotic lower bound for the stability of fractional Sobolev inequality: \begin{equation}\label{Sob sta ine} \left\|(-\Delta)^{s/2} U \right\|_2^2 - \mathcal S_{s,n} \| U\|_{\frac{2n}{n-2s}}^2\geq C_{n,s} d^{2}(U,…

In this paper, we prove the following reversed Hardy-Littlewood-Sobolev inequality with extended kernel \begin{equation*} \int_{\mathbb{R}_+^n}\int_{\partial\mathbb{R}^n_+} \frac{x_n^\beta}{|x-y|^{n-\alpha}}f(y)g(x) dydx\geq…

In this paper, we study the Cauchy problem for the inhomogeneous biharmonic nonlinear Schr\"{o}dinger (IBNLS) equation \[iu_{t} +\Delta^{2} u=\lambda |x|^{-b}|u|^{\sigma}u,~u(0)=u_{0} \in H^{s} (\mathbb R^{d}),\] where $d\in \mathbb N$,…

Given a smooth closed Riemannian manifold $(M,g)$ of dimension $N \ge 3$, we derive sharp quantitative stability estimates for nonnegative functions near the solution set of the Yamabe problem on $(M,g)$. The seminal work of Struwe (1984)…

In this paper, we study the stability of two inverse boundary value problems in an infinite slab with partial data. These problems have been studied by Li and Uhlmann for the case of the Schrodinger equation and by Krupchyk, Lassas and…

The Laplacian $\Delta_{\mathbb{S}^{n-1}}$ on the unit sphere $\mathbb{S}^{n-1}\subset \mathbb{R}^n$ has the property that it can explicitly be expressed in terms of $\Lambda$, the Dirichlet-to-Neumann map of the unit ball, as…

We prove that a local, weak Sobolev inequality implies a global Sobolev estimate using existence and regularity results for a family of $p$-Laplacian equations. Given $\Omega\subset\mathbb{R}^n$, let $\rho$ be a quasi-metric on $\Omega$,…

We prove the equivalence of Hardy- and Sobolev-type inequalities, certain uniform bounds on the heat kernel and some spectral regularity properties of the Neumann Laplacian associated with an arbitrary region of finite measure in Euclidean…

In this paper, we consider the following Caffarelli-Kohn-Nirenberg (CKN for short) inequality \begin{eqnarray*} \bigg(\int_{{\mathbb R}^d}|x|^{-b(p+1)}|u|^{p+1}dx\bigg)^{\frac{2}{p+1}}\leq S_{a,b}\int_{{\mathbb R}^d}|x|^{-2a}|\nabla u|^2dx,…

For a regular (in a sense) mapping $v:\mathbb{R}^n \to \mathbb{R}^d$ we study the following problem: {\sl let $S$ be a subset of $m$-critical a set $\tilde Z_{v,m}=\{{\rm rank} \nabla v\le m\}$ and the equality $\mathcal{H}^\tau(S)=0$ (or…

Inspired by a pioneer work of Andersson-Kapitanski \cite{AK}, we prove the local well-posedness of the Cauchy problem of incompressible neo-Hookean equations if the initial deformation and velocity belong to $H^{s+1}(\mathbb{R}^n) \times…

In this paper we continue investigation of the interior problem of tomography that was started in \cite{BKT2}. As is known, solving the interior problem {with prior data specified on a finite collection of intervals $I_i$} is equivalent to…

In this paper, we prove some a priori stability estimates (in weighted Sobolev spaces) for the spatially homogeneous Boltzmann equation without angular cutoff (covering every physical collision kernels). These estimates are conditioned to…

We study the nonlocal scalar field equation with a vanishing parameter \[ \left\{\begin{array}{lll} (-\Delta)^s u+\epsilon u &=|u|^{p-2}u -|u|^{q-2}u \quad\text{in}\quad\mathbb{R}^N \\ u >0, & u \in H^s(\mathbb{R}^N), \end{array} \right. \]…

Let $\Omega$ be a domain of $\mathbb R^n$ with $n\ge 2$ and $p(\cdot)$ be a local Lipschitz funcion in $\Omega$ with $1<p(x)<\infty$ in $\Omega$. We build up an interior quantitative second order Sobolev regularity for the normalized…