2024/04/26 by Ercole, Grey
#35B40 #35J92 #35J94 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2404.17103
Let Ω be a bounded, smooth domain of ℝN, N≥2. For p>N and 1≤ q(p)<∞ set λp,q(p):=inf\ ∫Ω\vert ∇ u\vert pdx:u∈ W01,p(Ω) and ∫Ω\vert u\vert q(p)dx=1\ and let up,q(p) denote a corresponding positive extremal function. We show that if limp→∞q(p)=∞, then limp→∞λp,q(p)1/p=\Vert dΩ\Vert ∞-1, where dΩ denotes the distance function to the boundary of Ω. Moreover, in the hyperdiffusive case: limp→∞(q(p))/(p)=∞, we prove that each sequence u_pn,q(pn), with pn→∞, admits a subsequence converging uniformly in Ω to a viscosity solution to the problem \ -Δ∞u=0 · amp; in · amp; Ω∖ M
u=0 · amp; on · amp; ∂Ω
u=1 · amp; in · amp; M, . where M is a closed subset of the set of all maximum points of dΩ.