On the validity of variational inequalities for obstacle problems with non-standard growth

Eleuteri, Michela; Napoli, Antonia Passarelli di

doi:10.54330/afm.114655

Cited by 9 publications

(3 citation statements)

References 14 publications

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“…We also assume that a solution to (1.1) is such that f (x, Du) ∈ L 1 loc (Ω). As it has been shown in [21], in case of non-standard growth condition (at least in the autonomous case), this turns to be the right class of competitors.…”

Section: Statement Of the Problem And Of The Main Resultsmentioning

confidence: 85%

Regularity for Obstacle Problems without Structure Conditions

Bertazzoni¹,

Riccò²

2021

Preprint

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This paper deals with the Lipschitz regularity of minimizers for a class of variational obstacle problems with possible occurance of the Lavrentiev phenomenon. In order to overcome this problem, the availment of the notions of relaxed functional and Lavrentiev gap are needed. The main tool used here is a ingenious Lemma which reveals to be crucial because it allows us to move from the variational obstacle problem to the relaxed-functional-related one. This is fundamental in order to find the solutions' regularity that we intended to study. We assume the same Sobolev regularity both for the gradient of the obstacle and for the coefficients.G. BERTAZZONI, S. RICC Ò So, the aim of this work is to complement the results cointained in the paper [11], where authors obtain Lipschitz regularity results for obstacle problems with Sobolev regularity for the coefficients and where the lagrangian f satisfies p, q−growth conditions without assuming that Lavrentiev phenomenon may occur. This phenomenon is a clear obstruction to regularity, since (1.2) prevents minimizers to belong to W 1,q . Notice that (1.2) cannot happen if p = q or if f is autonomous (it not depends on variable x) and convex. Moreover, as pointed out in Section 3 of [22], the appearance of (1.2) has geometrical reasons and cannot be spotted in a direct way by standard elliptic regularity techniques. Therefore, the basic strategy in getting regularity results consists in excluding the occurrence of (1.2) by imposing that the Lavrentiev gap functional L(u), defined in (2.10), vanishes on solutions. However, here in this manuscript we adopt a different viewpoint, following the lines of [20].We present a general Lipschitz regularity result by covering the case in which the Lavrentiev phenomenon may occur. In this respect, a key role will be played by the relaxed functional.We therefore need to introduce the exact framework of relaxation in the case of obstacle problems and then we will state our main result. The crucial step will be constituted by Lemma 5.1 which is the natural counterpart of the necessary and sufficient condition to get the absence of Lavrentiev phenomenon.We will state and prove the result with Sobolev dependence on both the obstacle and the partial map x → D ξ f (x, ξ). A model functional that is covered by our results isdx with q > p > 1 and a(•) a bounded Sobolev coefficient. The plan of the paper is the following: in Section 2 we state our model problem and the main results of the paper, in Section 3 we present some preliminary results we need in the sequel; Section 4 is devoted to the presentation of our a-priori estimate and finally in Section 5 we present the Lipschitz regularity results for solutions to the relaxed obstacle problem.

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Section: Statement Of the Problem And Of The Main Resultsmentioning

confidence: 85%

Regularity for Obstacle Problems without Structure Conditions

Bertazzoni¹,

Riccò²

2021

Preprint

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“…However, the embedding constant blows up as λ → ∞ unless ϕ also satisfies (aDec) p . This approximation approach is similar to that in [24]. Note in the next results that f is bounded by the Sobolev embedding in W 1,p (Ω).…”

Section: Minimizers With Non-doubling Growthmentioning

confidence: 74%

Bloch estimates in non-doubling generalized Orlicz spaces

Harjulehto,

Hästö,

Juusti

2022

Preprint

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We study minimizers of non-autonomous functionals inf u ˆΩ ϕ(x, |∇u|) dx when ϕ has generalized Orlicz growth. We consider the case where the upper growth rate of ϕ is unbounded and prove the Harnack inequality for minimizers. Our technique is based on "truncating" the function ϕ to approximate the minimizer and Harnack estimates with uniform constants via a Bloch estimate for the approximating minimizers.

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“…On the other hand, in the case of non-standard growth conditions, even in the unconstrained case, the relation between extremals and minima is an issue that requires a careful investigation (see for example [7,8,21]).…”

Section: Introductionmentioning

confidence: 99%

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

Grimaldi

Ipocoana

2022

Advances in Calculus of Variations

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We here establish the higher fractional differentiability for solutions to a class of obstacle problems with non-standard growth conditions. We deal with the case in which the solutions to the obstacle problems satisfy a variational inequality of the form ∫ Ω ⟨ A ⁢ ( x , D ⁢ u ) , D ⁢ ( φ - u ) ⟩ ⁢ d x ≥ 0 for all ⁢ φ ∈ K ψ ⁢ ( Ω ) , \int_{\Omega}\langle\mathcal{A}(x,Du),D(\varphi-u)\rangle\,dx\geq 0\quad\text{for all}\ \varphi\in\mathcal{K}_{\psi}(\Omega), where Ω is a bounded open subset of R n \mathbb{R}^{n} , ψ ∈ W 1 , p ⁢ ( Ω ) \psi\in W^{1,p}(\Omega) is a fixed function called obstacle and K ψ ⁢ ( Ω ) = { w ∈ W 1 , p ⁢ ( Ω ) : w ≥ ψ ⁢ a.e. in ⁢ Ω } \mathcal{K}_{\psi}(\Omega)=\{w\in W^{1,p}(\Omega):w\geq\psi\ \text{a.e. in}\ \Omega\} is the class of admissible functions. Assuming that the gradient of the obstacle belongs to some suitable Besov space, we are able to prove that some fractional differentiability property transfers to the gradient of the solution.

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On the validity of variational inequalities for obstacle problems with non-standard growth

Cited by 9 publications

References 14 publications

Regularity for Obstacle Problems without Structure Conditions

Regularity for Obstacle Problems without Structure Conditions

Bloch estimates in non-doubling generalized Orlicz spaces

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

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