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We consider solutions of p (x) -Laplacian systems with coefficients and we show that their gradient is continuous provided that the variable exponent has distributional gradient belonging to the Lorentz-Zygmund space L n , 1 log ⁡ L and that the gradient of the coefficient belongs to the Lorentz space L n , 1. The result is new since the use of the sharp Sobolev embedding in rearrangement invariant spaces does not ensure the unique (up to now) known assumption for such result, namely the log-Dini continuity of p (⋅) and the plain Dini continuity of the coefficient. Our approach relies on perturbation arguments and allows to slightly improve results in dimension two even for the case where p (⋅) is constant. [ABSTRACT FROM AUTHOR]

We prove C1regularity for local vectorial minimizers of the nonautonomous functional w ∈ W1,1loc (Ω; RN) 7→ Z Ω b(x) |Dw|p + a(x)|Dw|p log(e + |Dw|)dx, with Ω open subset of Rn, n ≥ 2, p > 1, 0 ≤ a(·) ≤ kakL∞(Ω) < ∞, and 0 < ν ≤ b(·) ≤ L. The result is valid provided that the function a(·) is logDini continuous and that the coefficient b(·) is Dini continuous or it is weakly differentiable and its gradient locally belongs to the Lorentz space Ln,1 (Ω; Rn). [ABSTRACT FROM AUTHOR]

Abstract We consider measure data problems of p -Laplacian type. The measure on the right-hand side has the property that the total variation of a generic ball decays in terms of generic functions of the radius; we show that this condition has a natural relation with gradient integrability properties and we get, as corollary, borderline cases of classic results. [ABSTRACT FROM AUTHOR]

We study the Cauchy-Dirichlet problem associated to a phase transition modeled upon the degenerate two-phase Stefan problem. We prove that weak solutions are continuous up to the parabolic boundary and quantify the continuity by deriving a modulus. As a byproduct, these a priori regularity results are used to prove the existence of a so-called physical solution. [ABSTRACT FROM AUTHOR]

We prove regularity results such as interior Lipschitz regularity and boundary continuity for the Cauchy–Dirichlet problem associated to a class of parabolic equations inspired by the evolutionary p -Laplacian, but extending it at a wide scale. We employ a regularization technique of viscosity-type that we find interesting in itself. [ABSTRACT FROM AUTHOR]

We consider solutions to singular parabolic equations with measurable dependence on the ( x , t ) variables and having on the right-hand side a measure satisfying a density condition. We prove that the less the measure is concentrated, the more the gradient is regular, in the Marcinkiewicz scale. We provide local estimates and recover some classic results. [ABSTRACT FROM AUTHOR]

We prove a Harnack inequality for minimisers of a class of non-autonomous functionals with non-standard growth conditions. They are characterised by the fact that their energy density switches between two types of different degenerate phases. [ABSTRACT FROM AUTHOR]

We consider solutions to nonlinear elliptic equations with measure data and general growth and ellipticity conditions of degenerate type, as considered in Lieberman (Commun Partial Differ Equ 16:311-361, ); we prove pointwise gradient bounds for solutions in terms of linear Riesz potentials. As a direct consequence, we get optimal conditions for the continuity of the gradient. [ABSTRACT FROM AUTHOR]

We consider solutions to degenerate parabolic equations with measurable coefficients, having on the right-hand side a measure satisfying a suitable density condition; we prove integrability results for the gradient in the Marcinkiewicz scale. [ABSTRACT FROM AUTHOR]

We derive the quantitative modulus of continuity which we conjecture to be optimal for solutions of the p-degenerate two-phase Stefan problem. Even in the classical case p = 2, this represents a twofold improvement with respect to the early 1980's state-of-the-art results by Caffarelli- Evans (Arch Rational Mech Anal 81(3):199-220, ) and DiBenedetto (Ann Mat Pura Appl 103(4):131-176, ), in the sense that we discard one logarithm iteration and obtain an explicit value for the exponent α( n, p). [ABSTRACT FROM AUTHOR]

We derive a class of potential estimates for elliptic equations with non-standard growth having a measure on the right-hand side; in particular our results allow to interpolate between the pointwise estimates available for solutions to this equation and the ones for the gradient. We allow bounded coefficients in BMO and VMO classes. [ABSTRACT FROM AUTHOR]

Abstract: We prove that the spatial gradient of (variational) solutions to parabolic obstacle problems of -Laplacian type enjoys the same regularity of the data and of the derivatives of the obstacle in the scale of Lorentz spaces. [Copyright &y& Elsevier]

Abstract: We prove estimates of Calderón–Zygmund type for evolutionary p-Laplacian systems in the setting of Lorentz spaces. We suppose the coefficients of the system to satisfy only a VMO condition with respect to the space variable. Our results hold true, mutatis mutandis, also for stationary p-Laplacian systems. [Copyright &y& Elsevier]

Abstract: Two processes are widely viewed as fundamental to the transition from conventional fossil-fuelled energy systems to renewable powered systems that is under way. There is firstly the progressive reduction in costs as investment, or production/energy generating capacity, grows. To see the uptake itself we need a second process, captured as a logistic curve (or S-shaped curve) that depicts the uptake of the new technology as an industrial substitution process unfolding over time. In this paper we put these two processes together, deriving a single expression that depicts uptake as a function of both cumulative investment and time, where the key parameter can be related to the learning coefficient. We display this expression in the form of a 3-dimensional surface that we dub the Logistic Industrial Surface. It is applied to a real case involving cost reduction and logistic uptake of solar PV (photovoltaic) cells. In this specific case, we estimate the learning curve involved and on this basis calculate that (for an initial time period) early in the trajectory a cost reduction of 8.7% would be associated with an increase in investment of 10%, leading to an increase in uptake by 4.35%; whereas a cost reduction of 44% (corresponding to a doubling of investment) would lead to a more rapid uptake of 41.95%. We claim that this is the first demonstration in the literature of a direct connection between investment levels, cost reductions and consequent levels of uptake according to logistic industrial dynamics. [Copyright &y& Elsevier]

Abstract: We consider parabolic equations of the type having a Radon measure on the right-hand side and prove fractional integrability and differentiability results of Calderón–Zygmund type for weak solutions. We extend some of the integrability results for elliptic equations achieved by G. Mingione (2007) to the parabolic setting and locally recover the integrability results of L. Boccardo, A. DallʼAglio, T. Gallouët, and L. Orsina (1997) in . [Copyright &y& Elsevier]

We consider a weak solution to the non-linear, parabolic systems of the form ut - div A( x, t, u, Du) = 0, where the vector field A satisfies a Dini-type continuity condition with respect to the variables ( x, t, u), and we prove a partial regularity result for such a solution. Moreover, we give an estimate of the size of the singular set of a solution in terms of a generalized parabolic Hausdorff measure associated to an appropriate modulus of continuity naturally associated to the coefficients of the system. [ABSTRACT FROM AUTHOR]

We prove sharp regularity results for a general class of functionals of the type w↦∫F(x,w,Dw)dx,featuring non-standard growth conditions and non-uniform ellipticity properties. The model case is given by the double phase integral w↦∫b(x,w)(|Dw|p+a(x)|Dw|q)dx,1with 0<ν≤b(·)≤L. This changes its ellipticity rate according to the geometry of the level set {a(x)=0} of the modulating coefficient a(·). We also present new methods and proofs that are suitable to build regularity theorems for larger classes of non-autonomous functionals. Finally, we disclose some new interpolation type effects that, as we conjecture, should draw a general phenomenon in the setting of non-uniformly elliptic problems. Such effects naturally connect with the Lavrentiev phenomenon. [ABSTRACT FROM AUTHOR]

Background: The role of body composition (lean mass and fat mass) on urine chemistries and bone quality is still debated. Our aim was therefore to determine the effect of lean mass and fat mass on urine composition and bone mineral density (BMD) in a cohort of healthy females.Materials and Methods: 78 female volunteers (mean age 46 ± 6 years) were enrolled at the Stone Clinic of Parma University Hospital and subdued to 24-hour urine collection for lithogenic risk profile, DEXA, and 3-day dietary diary. We defined two mathematical indexes derived from body composition measurement (index of lean mass-ILM, and index of fat mass-IFM) and the cohort was split using the median value of each index, obtaining groups differing only for lean or fat mass. We then analyzed differences in urine composition, dietary intakes and BMD.Results: The women with high values of ILM had significantly higher excretion of creatinine (991 ± 194 vs 1138 ± 191 mg/day, p = 0.001), potassium (47 ± 13 vs 60 ± 18 mEq/day, p < 0.001), phosphorus (520 ± 174 vs 665 ± 186 mg/day, p < 0.001), magnesium (66 ± 20 vs 85 ± 26 mg/day, p < 0.001), citrate (620 ± 178 vs 807 ± 323 mg/day, p = 0.002) and oxalate (21 ± 7 vs 27 ± 11 mg/day, p = 0.015) and a significantly better BMD values in limbs than other women with low values of ILM. The women with high values of IFM had similar urine composition to other women with low values of IFM, but significantly better BMD in axial sites. No differences in dietary habits were found in both analyses.Conclusions: Lean mass seems to significantly influence urine composition both in terms of lithogenesis promoters and inhibitors, while fat mass does not. Lean mass influences bone quality only in limb skeleton, while fat mass influences bone quality only in axial sites. [ABSTRACT FROM AUTHOR]