Higher differentiability for solutions of nonhomogeneous elliptic obstacle problems

Ma, Lingwei; Zhang, Zhenqiu

doi:10.1016/j.jmaa.2019.06.052

Cited by 11 publications

(6 citation statements)

References 20 publications

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“…Many recent works deal with regularity properties of solutions to variational problems in which the integrand depends on the x−variable trough a function that is possibly discontinuous, such as in the case of Sobolev-type dependence, under quadratic (see [34]), and super-quadratic growth conditions (see [19,20,21,22,28,33]). This kind of topics has been object of study also in the framework of obstacle problems (see [8,10,11,32]), even in the case of (p, q)-growth condition (such as in [7,14,15]).…”

Section: ˆωmentioning

confidence: 99%

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

Gentile¹

2020

Preprint

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We establish some higher differentiability results of integer and fractional order for solution to non-autonomous obstacle problems of the formwhere the function f satisfies p−growth conditions with respect to the gradient variable, for 1 < p < 2, and K ψ (Ω) is the class of admissible functions v ∈ u 0 + W 1,p 0 (Ω) such that v ≥ ψ a. e. in Ω, where u 0 ∈ W 1,p (Ω) 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) belongs to a suitable Sobolev or Besov space. The novelty here is that we deal with subquadratic growth conditions with respect to the gradient variable, i. e. f (x, ξ) ≈ a(x)|ξ| p with 1 < p < 2, and where the map a belongs to a Sobolev or Besov-Lipschitz space.

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Section: ˆωmentioning

confidence: 99%

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

Gentile¹

2020

Preprint

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“…More precisely, in [13] is proved the higher differentiability of the solution of an homogeneous obstacle problem with the energy density satisfying p-growth conditions; in [14] the integrand f depends also on the v variable; in [17] the energy density satisfies (p, q)-growth conditions. The nonhomogeneous obstacle problem is considered in [28] when the energy density satisfies p-growth conditions and in [7] when the energy density satisfies (p, q)-growth conditions. All previous quoted higher differentiability results have been obtained under a W 1,r with r ≤ n Sobolev assumption on the dependence on x of the operator A.…”

Section: ˆωmentioning

confidence: 99%

Regularity results for solutions to obstacle problems with Sobolev coeffcients

Caselli,

Gentile,

Giova

2019

Preprint

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We establish the higher differentiability of solutions to a class of obstacle problems of the typee. in Ω} and the convex integrand f satisfies p-growth conditions with respect to the gradient variable. We derive that the higher differentiability property of the weak solution v is related to the regularity of the assigned ψ, under a suitable Sobolev assumption on the partial map x → D ξ f (x, ξ). The main novelty is that such assumption is independent of the dimension n and that, in the case p ≤ n − 2, improves previous known results.

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“…The case of non-standard growth conditions has been faced, for example, in [21,22], for what concerns (p, q)-growth and in case of variable exponents in [18]. The case of non-homogeneous obstacle problems is faced in [35], where the energy density satisfies p-growth conditions, and in [7], where the energy density satisfies (p, q)-growth conditions. All previously quoted higher differentiability results have been obtained assuming that, with respect to the x-variable, the map A belongs to a Sobolev space W 1,r with r ≥ n. However, taking into account the result obtained for unconstrained problem in [11,30], proving that if we deal with bounded solutions to, the higher differentiability holds true under weaker assumptions on the partial map x → A(x, ξ) with respect to W 1,n , and the result obtained in [7] proving that a local bound assumption on the obstacle ψ implies a local bound for the solutions to the obstacle problem (1.1), in [8,26] have been proven that, if the obstacle is locally bounded, higher differentiability results for the solutions of (1.1) persist assuming that the partial map x → A(x, ξ) belongs to a Sobolev class that is not related to the dimension n but to the growth exponent p of the functional in case of standard growth and to the ellipticity and the growth exponents p and q of the functional in case of non-standard growth.…”

Section: Introductionmentioning

confidence: 99%

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

Gentile

2021

Forum Mathematicum

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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.

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Higher differentiability for solutions of nonhomogeneous elliptic obstacle problems

Cited by 11 publications

References 20 publications

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

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

Regularity results for solutions to obstacle problems with Sobolev coeffcients

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

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