Fractional differentiability for solutions of nonlinear elliptic equations

We study nonlinear elliptic equations in divergence form $${\operatorname{div}}{\mathcal A}(x,Du)={\operatorname{div}}G.$$ When ${\mathcal A}$ has linear growth in $Du$, and assuming that $x\mapsto{\mathcal A}(x,\xi)$ enjoys $B^\alpha_{\frac{n}\alpha, q}$ smoothness, local well-posedness is found in $B^\alpha_{p,q}$ for certain values of $p\in[2,\frac{n}{\alpha})$ and $q\in[1,\infty]$. In the particular case ${\mathcal A}(x,\xi)=A(x)\xi$, $G=0$ and $A\in B^\alpha_{\frac{n}\alpha,q}$, $1\leq q\leq\infty$, we obtain $Du\in B^\alpha_{p,q}$ for each $p<\frac{n}\alpha$. Our main tool in the proof is a more general result, that holds also if ${\mathcal A}$ has growth $s-1$ in $Du$, $2\leq s\leq n$, and asserts local well-posedness in $L^q$ for each $q>s$, provided that $x\mapsto{\mathcal A}(x,\xi)$ satisfies a locally uniform $VMO$ condition

A nutritional approach to the prevention of cancer: from assessment to personalized intervention

Among lifestyle factors, nutrition is one of the most important determinants of health, and represents a pivotal element of cancer risk. Nonetheless, epidemiological evidences of the relationship between several cancers and specific foods and nutrients is still inadequate, and solid conclusions are missing. Indeed, caloric restriction without malnutrition is associated to cancer prevention. Food may be also the primary route of exposure to contaminants such as metals, persistent organic pollutants, and pesticides. Exposuredisease associations and the interplay with genetic susceptibility requires further studies on genetic variation, environment, lifestyle, and chronic disease in order to eliminate and reduce associated health risks, thus contributing to improve health outcomes for the population. A primary nutritional approach for Active and Healthy Ageing (AHA) has been developed by the Nutrition group of the European Innovation Partnership (EIP) on AHA. The working group on lifestyles of the Italian Ministry of Health has developed a comprehensive approach to adequate nutrition using a consensus methodology to collect and integrate the available evidences from the literature and from the Italian experiences at the regional level, to raise the interest of other experts and relevant stakeholders to outline and scale-up joint strategies for a primary nutritional approach to cancer prevention

Quasiconformal mappings and exponentially integrable functions

We prove that a if-quasiconformal mapping f: R2 → R 2 which maps the unit disk D onto itself preserves the space EXP(D) of exponentially integrable functions over D, in the sense that u ∈ EXP(D) if and only if u o f-1 ∈ EXP(D). Moreover, if / is assumed to be conformal outside the unit disk and principal, we provide the estimate 1/1+K log K ≤ ||u o f-1||EXP(D)/||u||EXP(D)≤1+K log K for every u ε EXP(D). Similarly, we consider the distance from L∞ in EXP and we prove that if f: Ω → Ω' is a K-quasiconformal mapping and G ⊂ ⊂ Ω, then 1/K ≤ distEXP(f(G))(u o f-1, L∞(f(G)))/distEXP(f(G))(u, L∞(G)) ≤ K for every u ∈ EXP(G). We also prove that the last estimate is sharp, in the sense that there exist a quasiconformal mapping f: D → D, a domain G ⊂ ⊂ D and a function u ∈ EXP(G) such that distEXP(f(G))(u o f-1, L∞(f(G))) = K distEXP(f(G))(u, L∞(G))

Change of variables for A∞A_\infty weights by means of quasiconformal mappings: sharp results

Let f: Rn→Rn be a quasiconformal mapping whose Jacobian is denoted by Jf and let A∞ be the Muckenhoupt class of weights ω satisfying for every ball B ⊂ Rn and for some positive constant A ≥ 1 independent of B. We consider two characteristic constants ~ à (ω) and G̃1 (ω) which are simultaneously finite for every ω σ A∞. We study the behaviour of the Ã∞-constant under the operator already considered by Johnson and Neugebauer [18] and establish the equivalence of the two constants G̃1(Jf ) and Ã∞(Jf-1). Our quantitative esti-mates are sharp