Higher differentiability results in the scale of Besov spaces to a class of double-phase obstacle problems

2022/01/24 by Grimaldi, Antonio Giuseppe, Ipocoana, Erica
#26A27 #47J20 #49J40 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2201.09771

Abstract

We study the higher fractional differentiability properties of the gradient of the solutions to variational obstacle problems of the form min \biggl\ ∫_Ω F(x,w,Dw) d x : w ∈ K_ψ(Ω) \biggr\, with F double phase functional of the form F(x,w,z)=b(x,w)(|z|^p+a(x)|z|^q), where Ω is a bounded open subset of ℝⁿ, ψ∈ W^1,p(Ω) is a fixed function called obstacle and K_ψ(Ω)= \ w ∈ W^1,p(Ω) : w ≥ ψ a.e. in Ω\ is the class of admissible functions. Assuming that the gradient of the obstacle belongs to a suitable Besov space, we are able to prove that the gradient of the solution preserves some fractional differentiability property.

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