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混合问题与分数阶椭圆算子的谱结果

Spectral results for mixed problems and fractional elliptic operators

Gerd Grubb

备注：
21 pages, introduction expanded with more references, small improvements in formulations
发表信息：
J. Math. Anal. Appl. 421 (2015), 1616-1634
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摘要：

在第一部分的文件中，我们显示Weyl型谱渐近公式的伪微分算子$P_a$的顺序$2a$，与类型和因子分解指数$a\in R_+$，限制到紧集的边界，这包括分数幂的拉普拉斯算子。描述了本征函数的定义域和正则性。在第二部分中，我们将其应用于研究二阶强椭圆对称微分算子$A$在有界光滑集$Ω\subset R^n$上的混合问题的$A_{χ,Σ_+}$在$L_2(Ω)$中的实现;这里将边界$\partialΩ=Σ$光滑地划分为$Σ=Σ_-\cup Σ_+$，将Dirichlet条件$γ_0u=0$强加于$Σ_-$，并将Neumann或Robin条件$χu=0$强加于$Σ_+$。证明了Dirichlet-to-Neumann算子$P_{γ,χ}$主要是$\frac12$型的，其因子分解指数为$\frac12$，相对于$Σ_+$ .上述理论允许$D(A_{χ,Σ_+})$与$H^{\frac32}(Ω)$以外的奇异元素的详细描述，并导致一个频谱渐近公式的克莱因预解式差$A_{χ,Σ_+}^{-1}-A_γ^{-1}$。

英文摘要：

In the first part of the paper we show Weyl type spectral asymptotic formulas for pseudodifferential operators $P_a$ of order $2a$, with type and factorization index $a\in R_+$, restricted to compact sets with boundary; this includes fractional powers of the Laplace operator. The domain and the regularity of eigenfunctions is described. In the second part, we apply this in a study of realizations $A_{χ,Σ_+}$ in $L_2(Ω)$ of mixed problems for a second-order strongly elliptic symmetric differential operator $A$ on a bounded smooth set $Ω\subset R^n$; here the boundary $\partialΩ=Σ$ is partioned smoothly into $Σ=Σ_-\cup Σ_+$, the Dirichlet condition $γ_0u=0$ is imposed on $Σ_-$, and a Neumann or Robin condition $χu=0$ is imposed on $Σ_+$. It is shown that the Dirichlet-to-Neumann operator $P_{γ,χ}$ is principally of type $\frac12$ with factorization index $\frac12$, relative to $Σ_+$. The above theory allows a detailed description of $D(A_{χ,Σ_+})$ with singular elements outside of $H^{\frac32}(Ω)$, and leads to a spectral asymptotic formula for the Krein resolvent difference $A_{χ,Σ_+}^{-1}-A_γ^{-1}$.

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