Higher differentiability for bounded solutions to a class of obstacle problems with (p,q)-growth

2022/06/03 by Antonio Giuseppe Grimaldi, Grimaldi, Antonio Giuseppe · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Analytic and geometric function theory #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2206.01427

openalex publication_date 2022/06/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We establish the higher fractional differentiability of bounded minimizers to a class of obstacle problems with non-standard growth conditions of the form min \biggl\ ∫_Ω F(x,Dw)dx : w ∈ K_ψ(Ω) \biggr\, where Ω is a bounded open set of ℝⁿ, n ≥ 2, the function 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. If the obstacle ψ is locally bounded, we prove that the gradient of solution inherits some fractional differentiability property, assuming that both the gradient of the obstacle and the mapping x ↦ D_ξF(x,ξ) belong to some suitable Besov space. The main novelty is that such assumptions are not related to the dimension n.

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