Abstract
We prove two stability-type estimates involving the Schwarz rearrangement of the normalized first eigenfunction u 1 > 0 of certain linear elliptic operators whose first eigenvalue λ1 is close to the lowest possible one (i.e., \({\lambda_1^\star}\) , the first eigenvalue of the Dirichlet Laplacian in a suitable ball). In particular, we prove that if \({\lambda_1\approx \lambda_1^\star}\) then the L ∞-distance between the rearrangement \({u_1^\star}\) and the normalized first eigenfunction of the Dirichlet Laplacian corresponding to \({\lambda_1^\star}\) is less than a suitable power of the difference \({\lambda_1-\lambda_1^\star}\) times a universal constant. We also show that the L ∞-distance between the first eigenfunction of the Dirichlet Laplacian in a ball whose first eigenvalue equals λ1 and the rearrangement \({u_1^\star}\) can be controlled with a power of the value assumed by \({u_1^\star}\) on the boundary of that ball.
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Di Meglio, G. Some stability estimates for the symmetrized first eigenfunction of certain elliptic operators. Z. Angew. Math. Phys. 63, 835–853 (2012). https://doi.org/10.1007/s00033-012-0194-z
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DOI: https://doi.org/10.1007/s00033-012-0194-z