标题： 非光滑奇异格林算子的谱渐近性 显示英文标题

标题： Spectral asymptotics for nonsmooth singular Green operators

摘要： 奇异格林算子G通常作为椭圆边界值问题在区域\Omega \subset R^n上的预解式中的边界校正项出现，更一般地，它们出现在伪微分边界问题的演算中。 特别是，克雷因预解式公式中的边界项是一个奇异格林算子。 在光滑情况下已知，当G在有界区域上为负阶数-t时，其特征值或s-数的行为(*) s_j(G)\sim c j^{-t/(n-1)}对于j\to \infty ，由边界维数n-1决定。 在一些非光滑情况下，已知上界(**) s_j(G)\le Cj^{-t/(n-1)}。 我们证明当G是一个一般的自伴非负奇异Green算子，其符号在x中仅是Hölder连续时，(*)成立。 我们还证明了在比较强椭圆二阶微分算子（不一定是自伴的）的狄利克雷问题和一类诺伊曼问题的Krein预解公式中的边界项时，(*)对于t=2成立，该算子的系数属于W^1_p(\Omega )，其中p>n。 显示英文摘要

摘要： Singular Green operators G appear typically as boundary correction terms in resolvents for elliptic boundary value problems on a domain \Omega \subset R^n, and more generally they appear in the calculus of pseudodifferential boundary problems. In particular, the boundary term in a Krein resolvent formula is a singular Green operator. It is well-known in smooth cases that when G is of negative order -t on a bounded domain, its eigenvalues or s-numbers have the behavior (*) s_j(G) \sim c j^{-t/(n-1)} for j\to \infty, governed by the boundary dimension n-1. In some nonsmooth cases, upper estimates (**) s_j(G) \le Cj^{-t/(n-1)} are known. We show that (*) holds when G is a general selfadjoint nonnegative singular Green operator with symbol merely H\"older continuous in x. We also show (*) with t=2 for the boundary term in the Krein resolvent formula comparing the Dirichlet and a Neumann-type problem for a strongly elliptic second-order differential operator (not necessarily selfadjoint) with coefficients in W^1_p(\Omega) for some p>n.