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In this paper we consider a class of obstacle problems of the typewhere ψ is the obstacle, K ψ (Ω) = {v ∈ u0 + W 1,p 0 (Ω, R) : v ≥ ψ a.e. in Ω}, with u0 ∈ W 1,p (Ω) a fixed boundary datum, the class of the admissible functions and the integrand f (x, Dv) satisfies non standard (p, q)-growth conditions. We prove higher differentiability results for bounded solutions of the obstacle problem under dimension-free conditions on the gap between the growth and the ellipticity exponents. Moreover, also the Sobolev assumption on the partial map x → A(x, ξ) is independent of the dimension n and this, in some cases, allows us to manage coefficients in a Sobolev class below the critical one W 1,n .

In this paper, we consider a class of obstacle problems of the type $$\begin{aligned} \min \left\{ \int _{\Omega }f(x, Dv)\, {\mathrm d}x\,:\, v\in {\mathcal {K}}_\psi (\Omega )\right\} \end{aligned}$$ min ∫ Ω f ( x , D v ) d x : v ∈ K ψ ( Ω ) where $$\psi $$ ψ is the obstacle, $${\mathcal {K}}_\psi (\Omega )=\{v\in u_0+W^{1, p}_{0}(\Omega , {\mathbb {R}}): v\ge \psi \text { a.e. in }\Omega \}$$ K ψ ( Ω ) = { v ∈ u 0 + W 0 1 , p ( Ω , R ) : v ≥ ψ a.e. in Ω } , with $$u_0 \in W^{1,p}(\Omega )$$ u 0 ∈ W 1 , p ( Ω ) a fixed boundary datum, the class of the admissible functions and the integrand f(x, Dv) satisfies non standard (p, q)-growth conditions. We prove higher differentiability results for bounded solutions of the obstacle problem under dimension-free conditions on the gap between the growth and the ellipticity exponents. Moreover, also the Sobolev assumption on the partial map $$x\mapsto A(x, \xi )$$ x ↦ A ( x , ξ ) is independent of the dimension n and this, in some cases, allows us to manage coefficients in a Sobolev class below the critical one $$W^{1,n}$$ W 1 , n .

No abstract

We study the asymptotic behavior of a family of functionals which penalize a short-range interaction of convolution type between a finite perimeter set and its complement. We first compute the pointwise limit and we obtain a lower estimate on more regulars sets. Finally, some examples are discussed.