2019/07/18 by Li, Feng · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1907.08032
Let N≥ 1, s,k∈(0,1), p∈(1,∞). Let t>1, open bounded set Ω⊂\mathbb RN, R be the radius of Ω. Let BtR(Ω) be the ball containing Ω with radius tR and with the same center as Ω. In this article we study the asymptotic behavior of the first (s,p)-eigenvalue and corresponding first (s,p)-eigenfunctions during the approximation k→ s. We show that there exhibits a different phenomenon between the two directions of discontinuity of k→ s- and continuity of k→ s+, which can be triggered by behaviors of eigenfunctions on the boundary points bearing the positive Besov Capacity. And this difference prompts us to study the boundary behavior of operators (-Δp)s on the irregular boundary points. We also characterize some equivalent forms of the continuity case when k→ s-. In the end, we construct a counterexample for the discontinuity case during k→ s- based on the positivity of Besov capacity of Cantor set and the fine decay estimates up to the regular boundary points, used by P. Lindqvist and O. Martio. The proof works by reducing \widetilde Ws,p0(Ω) to the so-called Relative-nonlocal spaces \widetilde Ws,p0,tR(Ω) introduced here, which is equivalent to \widetilde Ws,p0(Ω), where \widetilde Ws,p0(Ω) is defined as the completion of C^∞0(Ω) under the Gagliardo semi-norm Ws,p(\mathbb RN), and \widetilde Ws,p0,tR(Ω) defined as the completion of C^∞0(Ω) under the Gagliardo semi-norm Ws,p(BtR(Ω)). As a partial result, we established the Homemorphism of the operator (-Δp)s between \widetilde Ws,p0(Ω) and its dual space \widetilde W-s,p^′(Ω), where 1/p+1/p^′=1.