Higher differentiability results for solutions to a class of non-autonomous obstacle problems with sub-quadratic growth conditions

Abstract

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.

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

“…In recent years, there has been a considerable interest in analyzing how an extra differentiability of integer or fractional order of the obstacle transfers to the gradient of solutions: for instance we quote [9,19,20,25,26,33] in the setting of standard growth conditions, [10,16,23,24,27,34,35,48] in the setting of non-standard growth conditions.…”

“…It is well known that no extra differentiability properties for the solutions can be expected even if the obstacle ψ is smooth, unless some assumption is given on the coefficients of the operator A. Therefore, recent results concerning the higher differentiability of solutions to obstacle problems show that a W 1,r Sobolev regularity, with r ≥ n, or a B s r,σ Besov regularity, with r ≥ n s , on the partial map x → A(x, ξ) is a sufficient condition (see [19,23,24,25] for the case of Sobolev class of integer order and [19,34] for the fractional one).…”

Higher differentiability for solutions to nonhomogeneous obstacle problems with 1<p<2

“…The aim of this article is to extend some higher differentiability results in [11] to non-homogeneous elliptic obstacle problems under the suitable conditions on the x-dependence of A. We make a suitable estimate of the nonhomogeneous term F by using the crucial Lemma 2.1 and other assumptions on A(x, ξ), thus obtaining new conclusions in Sobolev or Besov-Lipschitz space.…”

Regularity results for bounded solutions to obstacle problems with non-standard growth conditions

“…Here, as we already said, we are interested in higher differentiability results since in case of non standard growth, many questions are still open. In [6,7,12,17,18,21,27,31,38] the authors analyzed how an extra differentiability of integer or fractional order of the gradient of the obstacle provides an extra differentiability to the gradient of the solutions, also in case of standard growth. However, since no extra differentiability properties for the solutions can be expected even if the obstacle ψ is smooth, unless some assumption is given on the x-dependence of the operator A, the higher differentiability results for the solutions of systems or for the minimizers of functionals in the case of unconstrained problems (see [1,8,10,19,20,22,23,24,36,37]) have been useful and source of inspiration also for the constrained case.…”