Local Lipschitz continuity of minimizers with mild assumptions on the $x$-dependence

Eleuteri, Michela; Marcellini, Paolo; Mascolo, Elvira

doi:10.3934/dcdss.2019018

Cited by 5 publications

(5 citation statements)

References 22 publications

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“…In the last few years, the study of the regularity properties of solution and minimizers of problems with non standard growth conditions has undergone remarkable developments, motivated in part by the applications. We would mention the first papers of Marcellini [33,34,35] and the recent [36,37,38], the result on higher integrability and differentiability in [20,21,31,42] and more recently [3,4,9,10,11,14,15,16,17]. For complete details and references on problems with non standard growth we refer to the recent surveys [37,39] It is well known that a restriction between p and q is necessary by virtue of the celebrated counterexample by Marcellini (see [34]).…”

Section: The Novelty Of Theorem 11 Is In Two Directionsmentioning

confidence: 99%

“…In [16,17] for integral functionals under p, q, growth with Sobolev coefficients in L r , r > n, it has been shown that the bound q p < 1 + 1 n − 1 r permit to obatin the local Lipschitz continuity of the minimizers. Moreover, it is worth mentioning that the bound at (1.7) has been already used in [12] (see also [2]) for the study of problems with subquadratic non standard growth conditions.…”

Section: The Novelty Of Theorem 11 Is In Two Directionsmentioning

confidence: 99%

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Higher differentiability for a class of problems under p,q subquadratic growth

Mascolo¹,

Napoli²

2021

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Section: The Novelty Of Theorem 11 Is In Two Directionsmentioning

confidence: 99%

Section: The Novelty Of Theorem 11 Is In Two Directionsmentioning

confidence: 99%

Higher differentiability for a class of problems under p,q subquadratic growth

Mascolo¹,

Napoli²

2021

Preprint

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“…The relationship between the ellipticity and the growth exponent we impose, namely (2.2), is the one considered for the first time in the series of papers [17], [18], [19], [20] and it is sharp (in view of the well known counterexamples, see for instance [35]) also to obtain the Lipschitz continuity of solutions to elliptic equations and systems and minimizers of related functionals with p, q−growth. Regularity results under non standard growth conditions, a research branch started after the pioneering papers by Marcellini [36], [37], [38], has recently attracted growing attention, see among the others [2], [6], [12], [15], [32], [39], [40], [41].…”

Section: Introductionmentioning

confidence: 99%

Regularity for Obstacle Problems without Structure Conditions

Bertazzoni¹,

Riccò²

2021

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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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“…Sobolev dependence on the x variable. This is a recent joint research by Eleuteri-Marcellini-Mascolo [43], [44], [45], [46]. Previously we ware considering the variational differential equation…”

mentioning

confidence: 96%

“…not necessarily depending on the modulus of ξ of the type f (x, ξ) = g(x, |ξ|). We follow Eleuteri-Marcellini-Mascolo [43], [44], [45], [46]. We consider integrand without a prefixed structure condition…”

mentioning

confidence: 99%