A GEOMETRIC COVERING LEMMA AND NODAL SETS OF EIGENFUNCTIONS

作者：Han, Xiaolong^*; Lu, Guozhen

来源：Mathematical Research Letters, 2011, 18(2): 337-352.

• Covering lemma
• eigenfunctions
• nodal sets
• BMO estimates
• volume estimates
• Riemannian manifolds

The main purpose of this paper is two-fold. On one hand, we prove a sharper covering lemma in Euclidean space R-n for all n >= 2 (see Theorem 1.5). On the other hand, we apply this covering lemma to improve existing results for BMO and volume estimates of nodal sets for eigenfunctions u satisfying Delta u + lambda u = 0 on n-dimensional Riemannian manifolds when lambda is large (see Theorems 1.7, 1.8). We also improve the BMO estimates for the function q = vertical bar del(u)vertical bar(2) + lambda/nu(2) (see Theorem 1.10). Our covering lemma sharpens substantially earlier results and is fairly close to the optimal one we can expect (Conjecture 1.6).

• 出版日期2011-3
• 单位北京师范大学