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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, 1991); 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.

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.

We prove that the spatial gradient of (variational) solutions to parabolic obstacle problems of p-Laplacian type enjoys the same regularity of the data and of the derivatives of the obstacle in the scale of Lorentz spaces.

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.

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.

We prove local Calderon-Zygmund estimates for weak solutions of the evolutionary p(x, t)-Laplacian system partial derivative(t)u - div (a(x, t)vertical bar Du vertical bar(p(x,t)-2)Du) = div (vertical bar F vertical bar F-p(x,F-t)-2 under the classical hypothesis of logarithmic continuity for the variable exponent p(x, t). More precisely, we show that the spatial gradient Du of the solution is as integrable as the right-hand side F, i.e., vertical bar F vertical bar(p(.)) is an element of L-loc(q) double right arrow vertical bar Du vertical bar(p(.)) is an element of L-loc(q) for any q > 1, together with quantitative estimates. Thereby we allow the presence of eventually discontinuous coefficients a(x, t), requiring only a VMO condition with respect to the spatial variable x.

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.

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