A NATURAL LOWER BOUND FOR THE SIZE OF NODAL SETS

作者：Hezari Hamid^*; Sogge Christopher D

来源：Analysis & PDE, 2012, 5(5): 1133-1137.

DOI：10.2140/apde.2012.5.1133

• eigenfunctions
• nodal lines

We prove that, for an n-dimensional compact Riemannian manifold (M, g), the (n - 1)-dimensional Hausdorff measure vertical bar Z(lambda)vertical bar of the zero-set Z(lambda) of an eigenfunction e(lambda) of the Laplacian having eigenvalue -lambda, where lambda >= 1, and normalized by integral(M) vertical bar e(lambda)vertical bar(2)dV(g) = 1 satisfies
C vertical bar Z(lambda)vertical bar >= lambda(1/2) (integral(M) vertical bar e(lambda)vertical bar dV(g))(2)
for some uniform constant C. As a consequence, we recover the lower bound vertical bar Z(lambda)vertical bar >= lambda((3-n)/4).

• 出版日期2012