Yi Ru-Ya Zhang

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math.AP2026

Serrin's overdetermined theorem within Lipschitz domains

Hongjie Dong, Yi Ru-Ya Zhang

Let $I^c ◯ \subset R^{n}$ be a Lipschitz domain. We prove that, $I^c ◯$ satisfies the following Serrin-type overdetermined system $$u \in W^{1,2}(\mathbb R^n), \quad u=0\ \text{ a.e.…

math.AP2025

Sharp stability for critical points of the Sobolev inequality in the absence of bubbling

Gemei Liu, Yi Ru-Ya Zhang

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|^…

math.AP2025

Yi Ru-Ya Zhang

Let $I^c ◯ \subset R^{2}$ be a bounded domain and $u \in S B V (I^c ◯)$ be a local minimizer of the Mumford--Shah problem in the plane, with $0 \in S_{u}$ being a tip point and…

Serrin's overdetermined theorem within Lipschitz domains

Sharp stability for critical points of the Sobolev inequality in the absence of bubbling

A growth estimate for the planar Mumford--Shah minimizers at a tip point: An alternative proof of David--LÃ©ger