Some stability estimates for the symmetrized first eigenfunction of certain elliptic operators |
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Authors: | Guglielmo Di Meglio |
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Institution: | 1. Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, Complesso universitario Monte Sant’Angelo, v. Cintia, 80126, Napoli, Italy
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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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