Harmonic majorants for eigenfunctions of the Laplacian with finite Dirichlet integrals (Q1856887)

The author considers the Laplace operator on bounded domains \(\Omega\subset\mathbb R^N, N\geq 2\) with \(C^{1,1}\)-boundary. The main result of the paper concerns eigenfunctions \(\Delta f = \lambda f\) of the Laplacian with \(\lambda \geq 0\): if the Dirichlet integral \[ D_\alpha(f) = \int_\Omega \delta(x)^\alpha |\nabla f(x)|^2 dx < \infty \] for some \(\alpha\in(-1,1]\) (resp.\ \(\alpha\in (0,1]\) if \(N=2\)), then the function \(|f|^p\) with \(p=2(N-1)/(N+\alpha -2)\) admits a harmonic majorant on \(\Omega\). Conversely, if \(|f |^p\) has a harmonic majorant on \(\Omega\) for some \(p\in (1,2]\), then \(D_\alpha(f)<\infty\) for \(p\alpha = 2(N-1)-p(N-2)\). This result extends an earlier theorem due to \textit{S. Yamashita} [Ill. J. Math. 25, 626-631 (1981; Zbl 0466.31002)].