带凸势的Dirichlet p-拉普拉斯算子的基本间隙：尖锐一维界与高维二分法

中文摘要 AI 辅助

我们研究带凸势的有界凸域上Dirichlet p-拉普拉斯算子的基本间隙。通过正则化和两点最大值原理，我们证明了正第一特征函数的对数凹性。对于N≥2，我们通过收缩光滑凸域确定了p=2处的尖锐转变：当1<p<2时间隙消失，当p=2时间隙保持为D⁻²量级，当p>2时间隙发散。对于p≥2及凸势，我们首先建立退化加权庞加莱不等式，该不等式给出Lᵖ-庞加莱不等式的定量稳定性估计，进而得到基本间隙的无维度界；对于零势，我们还获得了包含第一特征值和直径的增强间隙估计。我们还证明了p>2时直径归一化间隙极小化子的存在性，并表明当p趋近于2时它们会退化。最后，对于N=1，我们证明了对所有p>1和所有凸势都成立的尖锐不等式λ₂,p(I_D,V)−λ₁,p(I_D,V)≥(p−1)(2ᵖ−1)(πₚ/D)ᵖ，当且仅当势为常数时等号成立。

英文摘要

We study the fundamental gap of the Dirichlet $p$-Laplacian on bounded convex domains. We first prove the sharp one-dimensional estimate $Γ_p(I_D,V)\ge (p-1)(2^p-1)(π_p/D)^p$, $I_D=(-D/2,D/2)$, for all $1<p<\infty$ and convex potentials $V$, with equality exactly for constant $V$. An analysis of collapsing convex domains shows that $p=2$ is exceptional: for $p\ne2$, the higher-dimensional gap is not governed by the sharp one-dimensional constant. For $V\equiv0$, define $G_{p,N}:=\inf_{Ω\subset\mathbb R^N}\operatorname{diam}(Ω)^pΓ_p(Ω,0)$ over bounded convex domains. We prove $G_{2,N}=3π^2$ and $G_{p,N}=0$ for $1<p<2$, $N\ge2$, while for every $p>2$, $G_{p,N}>0$ and the infimum is attained. We obtain explicit dimension-dependent bounds and $G_{p,N}\asymp_p N^{p-2}$ as $N\to\infty$, while for fixed $N$, $\lim_{p\downarrow2}G_{p,N}=3π^2$ and $\lim_{p\to\infty}G_{p,N}^{1/p}=4$. We also characterize the collapse as $p\downarrow2$: minimizing domains degenerate, and every limiting normalized ground-state measure is supported on a unit segment and, up to rigid motions, equals $2\cos^2(πt)\,dt$ on $(-1/2,1/2)$. The proof establishes log-concavity of the first $p$-eigenfunction with convex potential, a degenerate weighted Poincaré inequality for the ground-state measure, and a lower gap bound in terms of the first eigenvalue and the associated weighted Poincaré constant. These estimates also give the compactness and equality analysis for the collapsing minimizers.

发表机构

• Fudan University（复旦大学）
• Brandenburg University of Technology Cottbus–Senftenberg（科特布斯-森夫滕贝格勃兰登堡工业大学）
• The University of Sydney（悉尼大学）

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