发表机构
Institute of Mathematics, University of Rostock（罗斯托克大学数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究在非局部Robin边界条件下，通过构造控制椭圆算子的正特征函数并得到其双侧界，结合Nash不等式与对偶性，证明了高维有界Lipschitz区域上热半群的超收缩性并给出短时间范数阶。
AI 中文摘要
我们研究了维度d>2的有界Lipschitz区域Ω⊂ℝᵈ上满足非局部Robin边界条件的热半群。边界算子B∈ℒ(L²(∂Ω))可能会破坏半群的正性。我们假设存在正算子C∈ℒ(L²(∂Ω))，使得对所有u∈L²(∂Ω)有|Bu|≤C|u|，且C1∈L^∞(∂Ω)。在此假设下，我们证明了对应一致椭圆算子生成的半群具有超收缩性，更准确地说，其从L²(Ω)到L^∞(Ω)的范数在短时间内的阶为t^(-d/4)。证明的核心步骤是构造一个控制型椭圆算子的正特征函数φ，使其在Ω上几乎处处满足0<δ≤φ≤M。其中上界通过幂截断与Sobolev指数迭代得到，下界通过与Neumann半群比较得到。该双侧估计给出了比较半群的L^∞界，再结合Nash不等式与对偶性即可推出超收缩性。
英文摘要
We study heat semigroups on bounded Lipschitz domains $Ω\subset \mathbb{R}^{d}$ with dimension $d>2$ under non-local Robin boundary conditions. The boundary operator $B \in \mathcal{L}( \mathrm{L}^{2}(\partialΩ))$ is allowed to destroy the positivity of the semigroup. We assume that there exists a positive operator $C \in \mathcal{L}(\mathrm{L}^{2}(\partialΩ))$ such that $$ |Bu| \leq C|u| \quad\text{for every }u\in \mathrm{L}^{2}(\partialΩ), \qquad C{\bf 1}\in \mathrm{L}^{\infty}(\partialΩ). $$ Under this assumption, we prove that the semigroup generated by the corresponding uniformly elliptic operator is ultracontractive. More precisely, its norm from $\mathrm{L}^{2}(Ω)$ to $\mathrm{L}^{\infty}(Ω)$ has short-time order $t^{-d/4}$. The main step is the construction of a positive eigenfunction $ϕ$ of a dominating elliptic operator such that $$ 0 \ < \ δ\ \leq \ ϕ\ \leq \ M $$ almost everywhere in $Ω$. The upper bound is obtained by power truncations and an iteration of Sobolev exponents. The lower bound follows by comparison with the Neumann semigroup. The two-sided estimate gives an $\mathrm{L}^{\infty}$-bound for the comparison semigroup. Nash's inequality and duality then yield ultracontractivity.
