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Article overview
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Nodal sets of Laplace eigenfunctions: estimates of the Hausdorff measure in dimension two and three | | Alexander Logunov
; Eugenia Malinnikova
; | | Date: |
9 May 2016 | | Abstract: | Let $Delta_M$ be the Laplace operator on a compact $n$-dimensional
Riemannian manifold without boundary. We study the zero sets of its
eigenfunctions $u:Delta u + lambda u =0$. In dimension $n=2$ we refine the
Donnelly-Fefferman estimate by showing that $H^1({u=0 })le
Clambda^{3/4-eta}$, $eta in (0,1/4)$. The proof employs the
Donnelli-Fefferman estimate and a combinatorial argument, which also gives a
lower (non-sharp) bound in dimension $n=3$: $H^2({u=0})ge clambda^alpha$,
$alpha in (0,1/2)$. The positive constants $c,C$ depend on the manifold,
$alpha$ and $eta$ are universal. | | Source: | arXiv, 1605.2595 | | Services: | Forum | Review | PDF | Favorites | |
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