2021/07/07 by Allen, Mark, Kriventsov, Dennis, Neumayer, Robin
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2107.03495
For a domain Ω⊂ ℝn and a small number \frakT > 0, let E0(Ω) = λ1(Ω) + \frakT tor(Ω) = inf_u, w ∈ H10(Ω)∖ \0\ (∫ |∇ u|2)/(∫ u2) + \frakT ∫ (1)/(2) |∇ w|2 - w be a modification of the first Dirichlet eigenvalue of Ω. It is well-known that over all Ω with a given volume, the only sets attaining the infimum of E0 are balls BR; this is the Faber-Krahn inequality. The main result of this paper is that, if for all Ω with the same volume and barycenter as BR and whose boundaries are parametrized as small C2 normal graphs over ∂ BR with bounded C2 norm, ∫ |uΩ - uBR|2 + |Ω\triangle BR|2 ≤ C [E0(Ω) - E0(BR)] (i.e. the Faber-Krahn inequality is linearly stable), then the same is true for any Ω with the same volume and barycenter as BR without any smoothness assumptions (i.e. it is nonlinearly stable). Here uΩ stands for an L2-normalized first Dirichlet eigenfunction of Ω. Related results are shown for Riemannian manifolds. The proof is based on a detailed analysis of some critical perturbations of Bernoulli-type free boundary problems. The topic of when linear stability is valid, as well as some applications, are considered in a companion paper.