In this paper we prove the following result. Let $\Omega\subset\mathbb R^n, n\geq 3,$ be a bounded, strictly convex, smooth domain and $\varphi: \partial\Omega\rightarrow\mathbb R$ be a smooth function. Then for any $z\in\Omega,$ there exists $c_1=c_1(\Omega, \{z\}, n, \varphi)>0,$ such that if $c\geq c_1$ then the problem: $\sigma_{n-1}(D^2 u)=0$ in $\bar\Omega\setminus\{z\},$ $u|_{\partial\Omega}=\varphi,$ $\lim\limits_{r\rightarrow 0}\sup\limits_{B_r(z)}\frac{u(x)-u(z)}{|x-z|^{(n-2)/(n-1)}}=c,$ admits a smooth solution in $\bar\Omega\setminus\{z\}.$ Moreover, we obtain the optimal a priori estimates for this solution. In particular, we show for any integer $m\geq 0$ there exists $C_m=C_m(m, \Omega, \{z\}, n, \varphi, c)>0$ such that $|D^m u(x)|<C_m|x-z|^{\frac{n-2}{n-1}-m}.$ This work provides the first result demonstrating the existence of smooth solutions to the Dirichlet problem for fully nonlinear elliptic equations with prescribed density in Euclidean space. Previously, only continuous solutions were obtained.
Let $N\ge1$, $p\in[1,\infty)$, $\gamma\in(0,\infty)$, and $\Omega\subset\mathbb R^N$ be a bounded open interval when $N=1$ or a bounded Lipschitz domain when $N\ge2$. For any $\lambda\in(0,\infty)$ and any measurable function $u$, consider the weak-type nonlocal functional \begin{align*} G_{\lambda,p,\gamma}(u;\Omega)...
Xiao-Sheng Lin, Da-Chun Yang, Sibei Yang et al.· 2 citations· ⚡2
Given $\rho>0$, we consider the problem \[ \text{find $(\lambda,u) \in \mathbb{R} \times H_0^1(B)$ such that } \begin{cases} -\Delta u+\lambda u = |u|^{p-1}u&\text{in } B \\ \int_B u^2\,dx = \rho, \end{cases} \] where $B$ is a ball in $\mathbb{R}^N$, $N \ge 1$, and $1<p<2^*-1$. Without any further restriction on $N$, $...
Let $\Omega\subset\mathbb{R}^d$, $d\ge2$, be a bounded connected domain with boundary of class $C^{1,\alpha}$, where $0<\alpha<1$. For the adjoint Neumann--Poincar\'e operator $K^*_{\partial\Omega}$, normalised so that its distinguished eigenvalue is $1/2$, let $\lambda_j^+(\Omega)$ denote the upper min--max values on...
Let $n\ge3$, $\Omega\subset\mathbb R^n$ be an open set, $F:=\mathbb R^n\setminus\Omega$, and $\alpha\in(0,\infty)$. For any $x\in\Omega$, we define the capacitary distance \begin{align*} d_\alpha(x) := \inf\left\{ r>0: \operatorname{cap}(\overline{F\cap B(x,r)}) \ge \alpha\operatorname{cap}(B(\mathbf0,r)) \right\}. \en...
Yi-Qun Chen, Jie Xiao, Dachun Yang et al.· 0 citations
We consider the natural family of Fourier inequalities for the Paley--Wiener space $\mathrm{PW}^q(\Omega)$, consisting of $L^q$-functions with Fourier support in a convex set $\Omega \subset \mathbb{R}^n$, $n \geq 2$, free of affine lines. Namely, \[ \int_{\Omega}\dfrac{|\hat{f}(x)|^p}{\omega_{\Omega}^d(x)}dx\leq C\|f\...
Let $\Omega\subset\R^N$, $N\ge2$, be a bounded domain and let $1<p<\infty$. Inspired by the exponential case treated in \cite{BO2026}, we study the quasilinear Schr\"odinger--Maxwell system \[ \begin{cases} -\Delta_p u+\psi G'(u)=f&\text{in }\Omega,\\ -\Delta_p\psi=G(u)&\text{in }\Omega,\\ u=\psi=0&\text{on }\partial\O...
Genival da Silva· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
