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

Giuseppe Grimaldi, Antonio; Ipocoana, Erica

ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 51

We study the higher fractional differentiability properties of the gradient of the solutions to variational obstacle problems of the form

min\left\{{\int }_{\Omega }F\left(x,w,Dw\right)\mathrm{d}x:w\in {\kappa }_{\psi }\left(\Omega \right)\right\},

with F double phase functional of the form

F\left(x,w,z\right)=b\left(x,w\right)\left({\left|z\right|}^p+a\left(x\right){\left|z\right|}^q\right),

where Ω is a bounded open subset of ℝⁿ, ψ ∈ W^1,p(Ω) is a fixed function called obstacle and = {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.

Classification : 26A27, 49J40, 47J20
Keywords: Besov spaces, higher differentiability, obstacle problem, double phase, non-standard growth

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