Related papers: Sharp Liouville Theorems

A necessary and sufficient condition is established for an asymptotically stable renewal process to satisfy the strong renewal theorem. This result is valid for all alpha in (0, 1), thus completing a result for alpha in (1/2, 1) which was…

We extend Struwe's result (Acta Math., 1988) on the existence of free boundary constant mean curvature disks to almost every prescribed boundary contact angle in $(0, \pi)$. Specifically, let $\Sigma$ be a surface in $\mathbb{R}^3$…

The formation of singularity and breakdown of strong solutions to the two-dimensional (2D) Cauchy problem of the full compressible Navier-Stokes equations with zero heat conduction are considered. It is shown that for the initial density…

We establish the H\"older continuity of bounded nonnegative weak solutions to \begin{align*} \big(\Phi^{-1}(w)\big)_t=\Delta w+\nabla\cdot\big(a(x,t)\Phi^{-1}(w)\big)+b\big(x,t,\Phi^{-1}(w)\big), \end{align*} with convex $\Phi\in…

In this paper, we investigate the volume comparison theorem related to $\sigma_2$-curvature. In particular, we show that volume comparison theorem with respect to $\sigma_2$-curvature holds for metrics close to strictly stable positive…

We prove that bounded-degree expanders with non-negative Ollivier-Ricci curvature do not exist, thereby solving a long-standing open problem suggested by Naor and Milman and publicized by Ollivier (2010). In fact, this remains true even if…

It is known that for a possibly degenerate hypoelliptic Ornstein-Uhlenbeck operator $$ L= \frac{1}{2}\text{ tr} (QD^2 ) + \langle Ax, D \rangle = \frac{1}{2}\text{ div} (Q D ) + \langle Ax, D \rangle,\;\; x \in R^N, $$ all (globally)…

For an integral $2$-varifold $V=\underline{v}(\Sigma,\theta_{\ge 1})$ in $\mathbb{R}^n$ with generalized mean curvature $H\in L^2$ such that $\mu(\mathbb{R}^n)=4\pi$ and $\int_{\Sigma}|H|^2d\mu\le 16\pi(1+\delta^2)$ , we show that $\Sigma$…

In this note, we establish sharp regularity for solutions to the following generalized $p$- Poisson equation $$-\ div\ \big(\langle A\nabla u,\nabla u\rangle^{\frac{p-2}{2}}A\nabla u\big)=-\ div\ \mathbf{h}+f$$ in the plane (i.e. in…

In this paper we study the initial boundary value problem for the system $\mbox{div}(\sigma(u)\nabla\varphi)=0,\ \ u_t-\Delta u=\sigma(u)|\nabla\varphi|^2$. This problem is known as the thermistor problem which models the electrical heating…

Let $X=\{ X_n\}_{n\in \mathbb{Z}}$ be zero-mean stationary Gaussian sequence of random variables with covariance function $\rho$ satisfying $\rho(0)=1$. Let $\varphi:\mathbb{R}\to\mathbb{R}$ be a function such that…

Assume that $p\in[1,\infty]$ and $u=P_{h}[\phi]$, where $\phi\in L^{p}(\mathbb{S}^{n-1},\mathbb{R}^n)$ and $u(0) = 0$. Then we obtain the sharp inequality $|u(x)|\le G_p(|x|)\|\phi\|_{L^{p}}$ for some smooth function $G_p$ vanishing at $0$.…

Let $G$ be a locally compact abelian group, and let $\widehat{G}$ denote its dual group, equipped with a Haar measure. A variant of the uncertainty principle states that for any $S \subset G$ and $\Sigma \subset \widehat{G}$, there exists a…

Let $\Sigma$ be a smooth closed hypersurface with non-negative Ricci curvature, isometrically immersed in a space form. It has been proved in \cite{P}, \cite{CZ}, and \cite{C2} that there are some $L^2$ inequalities on $\Sigma$ which…

In this paper we prove weak L^{1,p} (and thus C^{\alpha}) compactness for the class of uniformly mean-convex Riemannian n-manifolds with boundary satisfying bounds on curvature quantities, diameter, and (n-1)-volume of the boundary. We…

We establish new Liouville-type theorems for the stationary Navier-Stokes equations in $\mathbb{R}^3$. A central open problem in this context is whether the classical $L^{9/2}(\mathbb{R}^3)$ condition of G.Galdi can be relaxed. In this note…

Smooth solutions to the axially symmetric Navier-Stokes equations obey the following maximum principle:$\|ru_\theta(r,z,t)\|_{L^\infty}\leq\|ru_\theta(r,z,0)\|_{L^\infty}.$ We first prove the global regularity of solutions if…

We consider a class of equations in divergence form with a singular/degenerate weight $$-\mathrm{div}(|y|^a A(x,y)\nabla u)=|y|^a f(x,y)\; \quad\textrm{or} \ \textrm{div}(|y|^aF(x,y))\;.$$ Under suitable regularity assumptions for the…

Let $\nabla$ be a metric connection with totally skew-symmetric torsion $\T$ on a Riemannian manifold. Given a spinor field $\Psi$ and a dilaton function $\Phi$, the basic equations in type II B string theory are \bdm \nabla \Psi = 0, \quad…

When $u$ is close to a single Talenti bubble $v$ of the $p$-Sobolev inequality, we show that \begin{equation*} \|Du-Dv\|_{L^p(\mathbb{R}^n)}^{\max\{1,p-1\}}\le C \|-{\rm div}(|Du|^{p-2}Du)-|u|^{p^*-2}u\|_{W^{-1,q}(\mathbb{R}^n)},…