Asymptotic behavior of normal derivatives of eigenfunctions for the Dirichlet Laplacian

We give answers to the problem posed by Ozawa in [S. Ozawa, Asymptotic property of eigenfunctions of the Laplacian at the boundary, Osaka J. Math. 30 (1993) 303–314]. For the Dirichlet Laplacian in a bounded domain, we define the function E(λ,x) from the normal derivatives, at a boundary point x, of the eigenfunctions whose corresponding eigenvalues do not exceed λ. If the domain is a ball, we show that Ozawaʼs conjecture is true, namely that E(λ,x) satisfies a two-term asymptotic formula as λ→∞. For a general C2 bounded domain, we improve the remainder estimate in the one-term asymptotic formula for E(λ,x), which Ozawa obtained.