2017/07/26 by Costea, Serban · 1 citation
#31C15 #46E35 (primary) #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1707.08873
This paper studies the Sobolev-Lorentz capacity and its regularity in the Euclidean setting for n ≥ 1 integer. We extend here our previous results on the Sobolev-Lorentz capacity obtained for n ≥ 2. Moreover, for n ≥ 2 integer we obtain a few new results concerning the n,1 relative and global capacities. We obtain sharp estimates for the n,1 relative capacity of the concentric condensers (B(0,r), B(0,1)) for all r in [0,1). As a consequence we obtain the exact value of the n,1 capacity of a point relative to all its bounded open neighborhoods from Rn when n ≥ 2. We also show that this aforementioned constant is the value of the n,1 global capacity of any point from Rn, where n ≥ 2 is integer. This allows us to give a new proof of the embedding H01,(n,1)(Ω) \hookrightarrow C(Ω) ∩ L∞(Ω), where Ω⊂ Rn is open and n ≥ 2 is an integer. In the penultimate section of our paper we prove a new weak convergence result for bounded sequences in the non-reflexive spaces H1,(p,1)(Ω) and H01,(p,1)(Ω). The weak convergence result concerning the spaces H1,(p,1)(Ω) is valid whenever 1