Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 44, 2019, 537-568

SOBOLEV–LORENTZ CAPACITY AND ITS REGULARITY IN THE EUCLIDEAN SETTING

Serban Costea

University of Pitesti, Department of Mathematics and Computer Science
Str. Targul din Vale nr. 1, RO-110040 Pitesti, Arges, Romania; serban.costea 'at' upit.ro

Abstract. 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. Specifically, we obtain sharp estimates for the n,1 relative capacity of the concentric condensers (\overline 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. These new sharp estimates concerning the n,1 relative capacity improve some of our previous results. We also obtain a new result concerning the n,1 global capacity. Namely, we show that this aforementioned constant is also the value of the n,1 global capacity of any point from Rⁿ, where n ≥ 2 is integer. Computing the aforementioned exact value of the n,1 relative capacity of a point with respect to all its bounded open neighborhoods from Rⁿ allows us to give a new prove of the embedding H₀^1,(n,1)(Ω) → C(\overline 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 H^1,(p,1)(Ω) and H₀^1,(p,1)(Ω). The weak convergence result concerning the spaces H^1,(p,1)(Ω) is valid whenever 1 < p < ∞, while the weak convergence result concerning the spaces H₀^1,(p,1)(Ω) is valid whenever 1 ≤ n < p < ∞ or 1 < n = p < ∞. As a consequence of the weak convergence result concerning the spaces H₀^1,(p,1)(Ω), in the last section of our paper we show that the relative and the global (p,1) and p,1 capacities are Choquet whenever 1 ≤ n < p < ∞ or 1 < n = p < ∞.

2010 Mathematics Subject Classification: Primary 31C15, 46E35.

Key words: Sobolev spaces, Lorentz spaces, capacity.

Reference to this article: S. Costea: Sobolev–Lorentz capacity and its regularity in the Euclidean setting. Ann. Acad. Sci. Fenn. Math. 44 (2019), 537-568.

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https://doi.org/10.5186/aasfm.2019.4433