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上的预解式的边界修正项出现，更一般地，它们出现在伪微分边界问题的计算中。 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，众所周知，当G在有界区域上为负阶-t时，其s-数具有行为(*) 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格林算子，其符号在 x 中仅是霍尔德连续时，(*) 成立。 我们还证明在比较强椭圆二阶微分算子（不一定自伴）的狄利克雷问题和一种诺伊曼型问题的克雷因预解公式中的边界项时，(*) 对 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 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.