N≥5及更高维下两类带临界无界系数椭圆问题的最优H²正则性
Sharp $H^2$-regularity in dimensions $N\geq 5$ and beyond for two classes of elliptic problems with critical unbounded coefficients
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中文总结 AI 辅助
该研究确定N≥5维下两类带临界无界系数椭圆问题的最优H²正则性参数阈值，扩展了前人结果，证明关键的二阶Hardy-Rellich型不等式。
中文摘要 AI 辅助
我们针对N≥5维下两类带临界无界扰动的椭圆问题，建立了控制H₀¹中弱解H²正则性的最优参数阈值。具体而言，我们考虑在包含原点x=0的有界C²区域Ω⊂ℝᴺ上提出的两类不同的λ参数椭圆问题：-Δv + λ(x·∇v)/|x|² = f 和 -Δv + λv/|x|² = f。我们注意到，奇异扰动项(x·∇v)/|x|²和v/|x|²是与拉普拉斯算子标度一致的2阶齐次算子。结合Hardy不等式，这两类问题分别在λ<(N-2)/2和λ>-(N-2)²/4时在H₀¹(Ω)中适定。主要结果如下：对于第一类问题，我们证明当f∈L²(Ω)时，任何属于H₀¹(Ω)的解v都属于H²(Ω)，这完全推广了Kim和Tsai在文献[Kim-Tsai]中针对λ≤0所得到的H²正则性结果；对于第二类问题，我们证明当λ>-N(N-4)/4时H²正则性成立，而当λ∈(-(N-2)²/4, -N(N-4)/4]时H²正则性不成立，这显著扩展了文献[Kato]中应用Kato扰动理论得到的λ∈(-N(N-4)/4, N(N-4)/4)的范围。此外，我们还为所涉及的椭圆算子建立了最优二阶Hardy-Rellich型不等式，这是上述证明的关键。
英文摘要
We establish sharp parameter thresholds governing $H^2$ -regularity of weak solutions in $H_0^1$ for two classes of elliptic problems with critical unbounded perturbations, in dimensions $N\geq 5$. More precisely, we consider two distinct $λ$-parametric elliptic problems $ - Δv + λ\frac{x\cdot \nabla v}{|x|^{2}} =f$ and $-Δv + λ\frac{v}{|x|^2}=f$ posed in a bounded $C^2$-domain $Ω\subset \mathbb{R}^N$ containing the origin $x=0$. We observe that the singular perturbations $\frac{x\cdot \nabla v}{|x|^{2}}$ and $\frac{v}{|x|^2}$ are homogeneous operators of order 2 consistent with the scaling of the Laplacian. In view of the Hardy inequality the problems are well-posed in $H_0^1(Ω)$ for $λ<\frac{N-2}{2}$ and $λ>-\frac{(N-2)^2}{4}$ respectively. The main results are as follows. For the first problem we show that any solution $v\in H_0^1(Ω)$ belongs to $H^2(Ω)$ for any $λ< \frac{N-2}{2}$ provided $f\in L^2(Ω)$. This fully extends the previous $H^2$ regularity properties obtained by Kim and Tsai in \cite{Kim-Tsai} for $λ\leq 0$. For the second problem we show that $H^2$ regularity holds for any $λ>- \frac{N(N-4)}{4}$ and fails for any $λ\in \left(-\frac{(N-2)^2}{4},-\frac{N(N-4)}{4}\right]$. This extends sharply the range of $λ\in \left(-\frac{N(N-4)}{4}, \frac{N(N-4)}{4}\right)$ obtained when applying the Kato perturbation theory in \cite{Kato}. In addition, we develop sharp second order Hardy-Rellich type inequalities for the involved elliptic operators which are essential in the above proofs.
