Higher fractional differentiability for solutions to a class of obstacle problems with non-standard growth conditions

2021/09/03 by Antonio Giuseppe Grimaldi, Grimaldi, Antonio Giuseppe, Erica Ipocoana +1
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2109.01584

Abstract

We here establish the higher fractional differentiability for solutions to a class of obstacle problems with non-standard growth conditions. We deal with the case in which the solutions to the obstacle problems satisfy a variational inequality of the form ∫_Ω ⟨ A(x,Du) ,D(φ-u) ⟩ dx ≥ 0 ∀ φ∈ K_ψ(Ω), 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 some suitable Besov space, we are able to prove that some fractional differentiability property transfers to the gradient of the solution.

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