Under a local L2 doubling condition, the nodal set of a solution to Δw = W·∇w + V w has (n−1)-dimensional Hausdorff measure bounded by a constant times a power of the Sobolev norms of W and V plus the doubling constant.
Nodal sets of Dirichlet eigenfunctions in quasiconvex Lipschitz domains
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abstract
We introduce the class of quasiconvex Lipschitz domains, which covers both $C^1$ and convex domains, to the study of boundary unique continuation for elliptic operators. In particular, we prove the upper bound of the size of nodal sets for Dirichlet eigenfunctions of general elliptic equations in bounded quasiconvex Lipschitz domains. Our result is new even for Laplace operator in convex domains.
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Nodal set for the Schr\"odinger equation under a local growth condition
Under a local L2 doubling condition, the nodal set of a solution to Δw = W·∇w + V w has (n−1)-dimensional Hausdorff measure bounded by a constant times a power of the Sobolev norms of W and V plus the doubling constant.