In the last decades, comparison results of Talenti type for Elliptic Problems with Dirichlet boundary conditions have been widely investigated. In this paper, we generalize the results obtained in Alvino et al. (Commun Pure Appl Math, to appear) to the case of p-Laplace operator with Robin boundary conditions. The point-wise comparison, obtained in Alvino et al. (to appear) only in the planar case, holds true in any dimension if p is sufficiently small.

No abstract

We consider functionals of the formwith convex integrand with respect to the gradient variable, assuming that the function that measures the oscillation of the integrand with respect to the x variable belongs to a suitable Sobolev space W 1,q . We prove a result of higer differentiability for the minimizers. We also infer a result of Lipschitz regularity of minimizers if q > n, and a result of higher integrability for the gradient if q = n. The novelty here is that we deal with integrands satisfying subquadratic growth conditions with respect to gradient variable. AMS Classifications. 49N60; 35J60; 49N99.

We establish some higher differentiability results of integer and fractional order for solutions to non-autonomous obstacle problems of the form min ⁡ { ∫ Ω f ⁢ ( x , D ⁢ v ⁢ ( x ) ) : v ∈ K ψ ⁢ ( Ω ) } , \min\biggl{\{}\int_{\Omega}f(x,Dv(x)):v\in\mathcal{K}_{\psi}(\Omega)\biggr{\}}, where the function 𝑓 satisfies 𝑝-growth conditions with respect to the gradient variable, for 1 < p < 2 1<p<2 , and K ψ ⁢ ( Ω ) \mathcal{K}_{\psi}(\Omega) is the class of admissible functions v ∈ u 0 + W 0 1 , p ⁢ ( Ω ) v\in u_{0}+W^{1,p}_{0}(\Omega) such that v ≥ ψ v\geq\psi a.e. in Ω, where u 0 ∈ W 1 , p ⁢ ( Ω ) u_{0}\in W^{1,p}(\Omega) is a fixed boundary datum. Here we show that a Sobolev or Besov–Lipschitz regularity assumption on the gradient of the obstacle 𝜓 transfers to the gradient of the solution, provided the partial map x ↦ D ξ ⁢ f ⁢ ( x , ξ ) x\mapsto D_{\xi}f(x,\xi) belongs to a suitable Sobolev or Besov space. The novelty here is that we deal with sub-quadratic growth conditions with respect to the gradient variable, i.e. f ⁢ ( x , ξ ) ≈ a ⁢ ( x ) ⁢ | ξ | p f(x,\xi)\approx a(x)\lvert\xi\rvert^{p} with 1 < p < 2 1<p<2 , and where the map 𝑎 belongs to a Sobolev or Besov–Lipschitz space.

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 .