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Doubling inequalities and critical sets of Dirichlet eigenfunctions. (English) Zbl 1469.35163

Summary: We study the sharp doubling inequalities for the gradients and upper bounds for the critical sets of Dirichlet eigenfunctions on the boundary and in the interior of compact Riemannian manifolds. Most efforts are devoted to obtaining the sharp doubling inequalities for the gradients. New idea is developed to overcome the difficulties on the unavailability of the double manifold in obtaining doubling inequalities in smooth manifolds. The sharp upper bounds of critical sets in analytic Riemannian manifolds are consequences of the doubling inequalities.

MSC:

35P20 Asymptotic distributions of eigenvalues in context of PDEs
35P15 Estimates of eigenvalues in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
35R01 PDEs on manifolds
58J05 Elliptic equations on manifolds, general theory
58C40 Spectral theory; eigenvalue problems on manifolds
28A78 Hausdorff and packing measures
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