arXiv subjects

Yabo Yang

Publications and source records attributed to Yabo Yang.

3 recordsLinked to original sources

arXiv2026

Anisotropic gradient rearrangement of BV functions and applications

In this paper, we introduce a symmetrization technique for the distributional gradient of a function of bounded variation in the anisotropic setting. This generalizes the result obtained in the Euclidean case in [Amato-Gentile-Nitsch-Trombetti, 2024] by separating the absolutely continuous part of the anisotropic gradient from its singular part. Our main result is an $L^1$ comparison between the function and its anisotropic symmetrization. Moreover, as an application, we derive isoperimetric inequalities for some geometric functionals related to the torsional rigidity.

math.AP↗

arXiv2024

Classification of positive solutions of critical anisotropic Sobolev equation without the finite volume constraint

In this paper, we classify all positive solutions of the critical anisotropic Sobolev equation \begin{equation}\label{0.1} -\Delta^{H}_{p}u = u^{p^{*}-1}, \ \ x\in \mathbb{R}^n \end{equation} without the finite volume constraint for $n \geq 3$ and $p_n(\Lambda) < p < n$, where $p^{*} = \frac{np}{n-p}$ denotes the critical Sobolev exponent, $-\Delta^{H}_{p}=-div(H^{p-1}(\cdot)\nabla H(\cdot))$ denotes the anisotropic $p$-Laplace operator and $\Lambda = \lambda\max\limits_{\substack{\xi \in \mathbb{R}^n\\1 \leq i, j \leq n}}\left\{\frac{|\xi|^{2}(\nabla^{2}_{ij}H^{p}(\xi))} {p(p-1)H^{p}(\xi)}\right\}$. By employing a novel approach based on invariant tensors technique, and using a Kato-type inequality, we prove that the positive solutions of \eqref{0.1} can be classified for $p_n(\Lambda) \leq p < n$, where $p_n(\Lambda)$ depends explicitly on $\Lambda$. This result removes the finite volume assumption on the classification of critical anisotropic $p$-Laplace equation which was obtained by Ciraolo-Figalli-Roncoroni in the literature \cite{CFR}. In particular, this results capture the precise dependence of critical exponents $p$ on both $n$ and $\Lambda$.

math.AP↗

arXiv2023

Classification of positive solutions of Hardy-Sobolev equation without the finite volume constraints

In this paper, we are concerned with the critical Hardy-Sobolev equation \begin{equation*} -\Delta_{p}u = \frac{u^{p^{*}_s-1}}{|x|^{s}}, \ \ x\in \mathbb{R}^n \end{equation*} where $p^{*}_s = \frac{(n-s)p}{n-p}$ denotes the critical Hardy-Sobolev exponent. We classify the positive solutions of this equation for $0 < s < \frac{p-1}{p}$ and $\frac{(2s+n+1)+\sqrt{(2s+n+1)^2-12s}}{6} \leq p < n$ without finite volume constraints, which extends Ou's result in \cite{9} in the literature. The method is based on constructing suitable vector fields integral inequality and using Newton's type inequality.

math.AP↗