1,721,001 research outputs found
Local boundedness of weak solutions to elliptic equations with p, q−growth
Full text linkThis article is dedicated to Giuseppe Mingione for his 50th birthday, a leading expert in the regularity theory and in particular in the subject of this manuscript. In this paper we give conditions for the local boundedness of weak solutions to a class of nonlinear elliptic partial differential equations in divergence form of the type considered below in (1.1), under p, q-growth assumptions. The novelties with respect to the mathematical literature on this topic are the general growth conditions and the explicit dependence of the differential equation on u, other than on its gradient Du and on the x variable
Regularity for Nonuniformly Elliptic Equations with p,q-Growth and Explicit x,u-Dependence
No full textWe are interested in the regularity of weak solutions u to the elliptic equation in divergence form; precisely in their local boundedness and their local Lipschitz continuity under general growth conditions, the so called p,q-growth conditions. We found a unique set of assumptions to get all these regularity properties at the same time; in the meantime we also found the way to treat a more general context, with explicit dependence on (x,u), other than on the gradient variable xi=Du; these aspects require particular attention due to the p,q-context, with some differences and new difficulties compared to the standard case p=q
Local lipschitz continuity of minimizers with mild assumptions on the x-dependence
No full textWe are interested in the regularity of local minimizers of energy integrals of the Calculus of Variations. Precisely, let Ω be an open subset of Rn. Let f (x, ξ) be a real function defined in Ω × Rnsatisfying the growth condition |fξx(x, ξ)| ≤ h (x) |ξ|p−1, for x ∈ Ω and ξ ∈ Rnwith |ξ| ≥ M0for some M0≥ 0, with h ∈ Lrloc(Ω) for some r > n. This growth condition is more general than those considered in the mathematical literature and allows us to handle some cases recently studied in similar contexts. We associate to f (x, ξ) the so-called natural p−growth conditions on the second derivatives fξξ(x, ξ); i.e., (p − 2) −growth for |fξξ(x, ξ)| from above and (p − 2) −growth from below for the quadratic form (fξξ(x, ξ) λ, λ); for details see either (3) or (7) below. We prove that these conditions are sufficient for the local Lipschitz continuity of any minimizer u ∈ Wloc1,p(Ω) of the energy integral fΩf (x, Du (x)) dx
Regularity for scalar integrals without structure conditions
Full text linkIntegrals of the Calculus of Variations with p, q-growth may have not smooth minimizers, not even bounded, for general p, q exponents. In this paper we consider the scalar case, which contrary to the vector-valued one allows us not to impose structure conditions on the integrand f (x, ξ) with dependence on the modulus of the gradient, i.e. f(x , ξ) = g (x,|ξ|). Without imposing structure conditions, we prove that if q p is sufficiently close to 1, then every minimizer is locally Lipschitz-continuous
CalculusofVariations. – TheSobolevclasswhereaweaksolutionisalocalminimizer, by Filomena De Filippis, Francesco Leonetti, Paolo Marcellini and Elvira Mascolo, communicated on 10 February 2023
Full text linkThe aim of this paper is to propose some results which we hope could contribute to understand better Lavrentiev’s phenomenon for energy integrals as in (1.1) under some p; q-growth conditions as in (1.2); in fact, we expect that Lavrentiev’s phenomenon does not occur if the quotient q=p is not too large in dependence of n, for instance, as in the cases – either scalar or vectorial ones – that we consider in this manuscript
Growth inhibition and induction of specific hepatic phenotype epression by retinoic acid in HepG2 cells
No full textRetinoic acid, the active metabolite of vitamin A, plays a role in the growth and differentiation of a variety of normal and malignant cells. In response to 5 microM retinoic acid the human hepatoma-derived cell line HepG2 underwent significant growth inhibition (not associated with cell death), which reached a level of 80% in comparison with controls, after 12 days of continuous treatment. Retinoic acid also induced morphological changes in these cells, in particular the development of canalicular-like structures, indicating progression to a more differentiated phenotype. In addition, a reduced expression of alpha-fetoprotein was found. We suggest that our results may be important for the design of novel therapeutic approaches using RA for the treatment of liver tumors
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