Jul 2026· Far East Journal of Mathematical Sciences (FJMS)· Vol 143, pp. 3171-3187· 0 citations
Abstract
We prove Radó theorem for nonlinear wave equations. Considering $u$ as a locally Lipschitz continuous function on an open set $\mathcal{X}$ in $\mathbb{R}^{n+1}$ to be a weak solution of the equations $ \partial_{tt}u-div(|u’|^{p-2}u’)=0$, on $\mathcal{X}\backslash u^{-1}(0)$, where $p \ge 2$, and employing the Hausdorff measure, we show that $u$ is a weak solution of these nonlinear wave equations in the entire $\mathcal{X}.$
This paper is concerned with the following semilinear elliptic equation involving the fractional Laplacian: $$(-\Delta)^s u+ g|u|^{p-1}u= \lambda \frac{u}{|x|^{2s}}+f(x),$$ in a bounded domain $\Omega$ of $\mathbb{R}^N\,(N>2s)$, subject to the zero Dirichlet condition in $\mathbb{R}^N\setminus \Omega$, where $01$ and $...
For continuous viscosity solutions $u$ of fully nonlinear uniformly elliptic equations $F(D^2 u)=0$, the work of Nadirashvili--Tkachev--Vl\u{a}du\c{t} shows that $u$ need not be $C^2$ in dimensions $n\ge 5$. It is therefore natural to ask how large the set $\operatorname{Sing}(u)$, consisting of points at which $u$ has...
Let $\mathbb H^n$ be the $n$-dimensional hyperbolic space, and $\Delta_{\mathbb H^n}$ be the Laplace--Beltrami operator on $\mathbb H^n$. Let $u=u(t,\cdot)$ be the solution to the wave equation $\partial^2_{t}u-\big(\Delta_{\mathbb H^n}+\frac{1}{4}(n-1)^2\operatorname{Id}\big)u=0$ with initial conditions $u(0,x)=f(x)$...
Peng Chen, X. Duong, Yun-Xiang Wang et al.· 0 citations
We prove the constant-one discrete nonlinear Hausdorff-Young inequality. As a consequence, by a discrete-to-continuous limiting argument, we obtain $$ \|(\log|a_f|^{2})^{1/2}\|_{L^{p'}(\mathbb{R})} \le \|f\|_{L^{p}(\mathbb{R})},\quad 1\le p<2, $$ for all $f\in L^{p}(\mathbb{R})$, where $a_f$ denotes the transmission co...
In this note, we prove uniform upper bounds for the volume of the level set $$\{x\in\Omega: c\le f(x)0,$$ for strictly convex and hyperbolic functions $f$ defined on a convex domain $\Omega\subset\mathbb{R}^n$ ($n\ge 2$). In particular, under a Hessian lower bound $D^2f\ge I_n$, we obtain the sharp volume bound $$|S^{n...
In this paper, it is proved that the commutator $\mathcal{H}_{β,b}$ which is generated by the $n$-dimensional fractional Hardy operator $\mathcal{H}_β$ and $b\in \dot{Λ}_α(\mathbb{R}^n)$ is bounded from $L^P(\mathbb{R}^n)$ to $L^q(\mathbb{R}^n)$, where $0
Qin Zheng, Zun-Wei Fu· Communications in Mathematic...· 1 citation
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