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Yi-Tian Zhang

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Preprint Jul 2026

Sharp Gradient Stability for the Sobolev Trace Inequality

Let $n\ge3$ and $1<p<n$. We prove a quantitative stability estimate for the critical Sobolev trace inequality on the upper half-space. More precisely, the Sobolev trace deficit controls the $\max\{2,p\}$-th power of the gradient distance to the manifold of trace bubbles. A central part of the proof is the spectral nondegeneracy of the trace bubbles: the first two eigenspaces of the linearized weighted Steklov problem are exactly the amplitude, dilation, and tangential translation modes.

Xi-Nan Ma, Yi-Tian Zhang, Yang Zhou · 1 citation · ⚡1

Preprint Aug 2026

The complete spectrum of the linearized $p$-Laplacian at a Sobolev extremal

Let $1<p<n$ and let $v(x)=(1+|x|^{p/(p-1)})^{-(n-p)/p}$ be the standard radial extremal for the sharp Sobolev inequality. We determine all eigenvalues and eigenspaces of the linearized $p$-Laplacian at $v$, defined by its closed quadratic form in $L^2(\mathbb{R}^n,v^{p^*-2} dx)$. After decomposition into spherical harmonics, an explicit gauge transformation and a change of variables identify each radial operator with a shifted Jacobi operator. This yields a complete eigenbasis indexed by $(\ell,k)\in\mathbb{N}_0^2$, where $\ell$ is the angular degree and $k$ is the radial mode number.

Yi-Tian Zhang · 0 citations