2019
DOI: 10.1515/forum-2019-0148
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Higher differentiability of solutions to a class of obstacle problems under non-standard growth conditions

Abstract: We establish the higher differentiability of integer order of solutions to a class of obstacle problems assuming that the gradient of the obstacle possesses an extra integer differentiability property. We deal with the case in which the solutions to the obstacle problems satisfy a variational inequality of the form\int_{\Omega}\langle\mathcal{A}(x,Du),D(\varphi-u)\rangle\,dx\geq 0\quad\text{% for all }\varphi\in\mathcal{K}_{\psi}(\Omega).The main novelty is that the operator {\mathcal{A}} satisfies the so-call… Show more

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Cited by 22 publications
(14 citation statements)
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“…Dealing with bounded solutions, we are able to prove our result 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 ellipticity and the growth exponents p and q of the functional and this assumption in case p+2 p−q+1 < n (i.e. p < n − 2 and q < n−1 n p + n−2 n ) improves the higher differentiability result obtained in [18]. Moreover, our result is obtained under a weaker assumption also on the gradient of the obstacle, indeed previous result assumed ψ ∈ W 1,2q−p (see [18]) while our hypothesis is ψ ∈ W 1, p+2 p+2−q , and under our assumption on the gap, i.e.…”
Section: Introductionsupporting
confidence: 65%
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“…Dealing with bounded solutions, we are able to prove our result 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 ellipticity and the growth exponents p and q of the functional and this assumption in case p+2 p−q+1 < n (i.e. p < n − 2 and q < n−1 n p + n−2 n ) improves the higher differentiability result obtained in [18]. Moreover, our result is obtained under a weaker assumption also on the gradient of the obstacle, indeed previous result assumed ψ ∈ W 1,2q−p (see [18]) while our hypothesis is ψ ∈ W 1, p+2 p+2−q , and under our assumption on the gap, i.e.…”
Section: Introductionsupporting
confidence: 65%
“…p < n − 2 and q < n−1 n p + n−2 n ) improves the higher differentiability result obtained in [18]. Moreover, our result is obtained under a weaker assumption also on the gradient of the obstacle, indeed previous result assumed ψ ∈ W 1,2q−p (see [18]) while our hypothesis is ψ ∈ W 1, p+2 p+2−q , and under our assumption on the gap, i.e. q − p < 1, it results W 1,2q−p ֒→ W 1, p+2 p+2−q .…”
Section: Introductionsupporting
confidence: 54%
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“…The regularity for solutions of obstacle problems has been object of intense study not only in the case of variational inequalities modelled upon the p-Laplacean energy [8,9,13,29] but also in the case of more general structures [4,5,11,16,17] It is usually observed that the regularity of solutions to the obstacle problems depends on the regularity of the obstacle itself: for linear problems the solutions are as regular as the obstacle; this is no longer the case in the nonlinear setting for general integrands without any specific structure. Hence along the years, in this situation there has been an intense research activity in which extra regularity has been imposed on the obstacle to balance the nonlinearity (see [2,3,9,15,16,27])…”
Section: ˆωmentioning
confidence: 99%
“…Here Ω is a bounded open set of R n , n > 2 and F : Ω × R n → R is a Carathéodory function fulfilling natural growth and convexity assumptions with variable exponent (namely assumptions (A1)-(A3) below). Higher diferentiability results have been attracting a lot of attention in the recent years, starting from the pioneering papers [32,33,34,21,22,29], to the more recent results concerning higher differentiability results for obstacle problems in the case of standard growth conditions [13,14,24] and p − q growth conditions of integer [17,16,6] and fractional order [25], see also [6], including the case of nearly linear growth [18] and the subquadratic growth case [19]. In the same spirit of these results, assuming that the gradient of the obstacle belongs to a suitable Sobolev class, we are interested in finding conditions on the partial map x −→ A(x, ξ) := D ξ F (x, ξ) in order to obtain that the extra differentiability property of the obstacle transfers to the gradient of the solution, possibly with no loss in the order of differentiability.…”
Section: Introductionmentioning
confidence: 99%