Abstract
We establish sharp upper bounds on the (n−1)-dimensional Hausdorff measure of the zero (nodal) sets and on the maximal order of vanishing corresponding to eigenfunctions of a regular elliptic problem on a bounded domain Ω ⊆ ℝn with real-analytic boundary. The elliptic operator may be of an arbitrary even order, and its coefficients are assumed to be real-analytic. This extends a result of Donnelly and Fefferman ([DF1], [DF3]) concerning upper bounds for nodal volumes of eigenfunctions corresponding to the Laplacian on compact Riemannian manifolds with boundary.
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Kukavica, I. Nodal volumes for eigenfunctions of analytic regular elliptic problems. J. Anal. Math. 67, 269–280 (1995). https://doi.org/10.1007/BF02787793
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DOI: https://doi.org/10.1007/BF02787793