论文标题
在非平滑案例中的分数dirichlet实现的Weyl渐近学
Weyl asymptotics for fractional-order Dirichlet realizations in nonsmooth cases
论文作者
论文摘要
令$ p $为对称$ 2A $ - 订单经典的强烈椭圆形伪数运算符，$ r^n $上的符号$ p（x，ξ）$（$ 0 <a <a <1 $），例如$（ - δ）^a $。令$ω\子集r^n $被界定，让$ p_d $为$ l_2（ω）$在外部条件下定义的$ u = 0 $ in $ r^n \setMinusΩ$中的dirichlet实现。当$ p（x，ξ）$和$ω$是$ c^\ infty $时，众所周知，特征值$λ_j$（以$ j \ to \ infty $的非额定顺序订购） λ_j（p_ {d}）= c（p，ω）j^{2a/n}+o（j^{2a/n}）\ text {for} j \ to} j \ to \ infty，$ c（p，p，ω）$从$ p $的主要符号确定。现在，我们表明该结果对于更可能是非平滑$ x $依赖性（在Lipschitz域）的更普通运营商有效，并且它扩展到$ \ tilde p = p = p+p+p+p'+p''$，其中$ p'$是订单的运营商$ <\ \ \ \ \米\ \米\ {2a，a+\ frac12 $ propalies and propalies and y Min \ frac12 $' $ l_2（ω）$（例如$ p''= v（x）\ in l_ \ infty（ω）$）。还讨论了$ p_d $的本征函数的规律性。
Let $P$ be a symmetric $2a$-order classical strongly elliptic pseudodifferential operator with even symbol $p(x,ξ)$ on $R^n$ ($0<a<1$), for example a perturbation of $(-Δ)^a$. Let $Ω\subset R^n$ be bounded, and let $P_D$ be the Dirichlet realization in $L_2(Ω)$ defined under the exterior condition $u=0$ in $R^n\setminusΩ$. When $p(x,ξ)$ and $Ω$ are $C^\infty $, it is known that the eigenvalues $λ_j$ (ordered in a nondecreasing sequence for $j\to\infty $) satisfy a Weyl asymptotic formula $$ λ_j(P_{D})=C(P,Ω)j^{2a/n}+o(j^{2a/n})\text{ for }j\to \infty, $$ with $C(P,Ω)$ determined from the principal symbol of $P$. We now show that this result is valid for more general operators with a possibly nonsmooth $x$-dependence, over Lipschitz domains, and that it extends to $\tilde P=P+P'+P''$, where $P'$ is an operator of order $<\min\{2a, a+\frac12\}$ with certain mapping properties, and $P''$ is bounded in $L_2(Ω)$ (e.g. $P''=V(x)\in L_\infty (Ω)$). Also the regularity of eigenfunctions of $P_D$ is discussed.
