Quantization for an elliptic equation of order 2m with critical exponential non-linearity

On a smoothly bounded domain $\Omega\subset\R{2m}$ we consider a sequence of positive solutions $u_k\stackrel{w}{\rightharpoondown} 0$ in $H^m(\Omega)$ to the equation $(-\Delta)^m u_k=\lambda_k u_k e^{mu_k^2}$ subject to Dirichlet boundary conditions, where $0<\lambda_k\to 0$. Assuming that $$\Lambda:=\lim_{k\to\infty}\int_\Omega u_k(-\Delta)^m u_k dx<\infty,$$ we prove that $\Lambda$ is an integer multiple of \Lambda_1:=(2m-1)!\vol(S^{2m}), the total $Q$-curvature of the standard $2m$-dimensional sphere.Comment: 33 page

Biomaterial-mediated factor delivery for spinal cord injury treatment

Spinal cord injury (SCI) is an injurious process that begins with immediate physical damage to the spinal cord and associated tissues during an acute traumatic event. However, the tissue damage expands in both intensity and volume in the subsequent subacute phase. At this stage, numerous events exacerbate the pathological condition, and therein lies the main cause of post-traumatic neural degeneration, which then ends with the chronic phase. In recent years, therapeutic interventions addressing different neurodegenerative mechanisms have been proposed, but have met with limited success when translated into clinical settings. The underlying reasons for this are that the pathogenesis of SCI is a continued multifactorial disease, and the treatment of only one factor is not sufficient to curb neural degeneration and resulting paralysis. Recent advances have led to the development of biomaterials aiming to promote in situ combinatorial strategies using drugs/biomolecules to achieve a maximized multitarget approach. This review provides an overview of single and combinatorial regenerative-factor-based treatments as well as potential delivery options to treat SCIs

Spherical harmonics and integration in superspace

In this paper the classical theory of spherical harmonics in R^m is extended to superspace using techniques from Clifford analysis. After defining a super-Laplace operator and studying some basic properties of polynomial null-solutions of this operator, a new type of integration over the supersphere is introduced by exploiting the formal equivalence with an old result of Pizzetti. This integral is then used to prove orthogonality of spherical harmonics of different degree, Green-like theorems and also an extension of the important Funk-Hecke theorem to superspace. Finally, this integration over the supersphere is used to define an integral over the whole superspace and it is proven that this is equivalent with the Berezin integral, thus providing a more sound definition of the Berezin integral.Comment: 22 pages, accepted for publication in J. Phys.

A threshold phenomenon for embeddings of H0mH^m_0 into Orlicz spaces

We consider a sequence of positive smooth critical points of the Adams-Moser-Trudinger embedding of $H^m_0$ into Orlicz spaces. We study its concentration-compactness behavior and show that if the sequence is not precompact, then the liminf of the $H^m_0$-norms of the functions is greater than or equal to a positive geometric constant.Comment: 14 Page

The right to food and food diversity in the Italian Constitution

Il contributo analizza la tutela apprestata dalla Costituzione italiana al diritto al cibo che, pur non essendo espressamente menzionato, viene ricavato attraverso l'analisi di principi ed azioni sottese alla nostra Carta che ne riconoscono il valore: il principio lavorista, la lotta alla povertà, la retribuzione del lavoratore...