Some remarks on the isoperimetric problem for the higher eigenvalues of the Robin and Wentzell Laplacians

We consider the problem of minimising the $k$th eigenvalue, $k \geq 2$, of the ($p$-)Laplacian with Robin boundary conditions with respect to all domains in $\mathbb{R}^N$ of given volume $M$. When $k=2$, we prove that the second eigenvalue of the $p$-Laplacian is minimised by the domain consisting of the disjoint union of two balls of equal volume, and that this is the unique domain with this property. For $p=2$ and $k \geq 3$, we prove that in many cases a minimiser cannot be independent of the value of the constant $\alpha$ in the boundary condition, or equivalently of the volume $M$. We obtain similar results for the Laplacian with generalised Wentzell boundary conditions $\Delta u + \beta \frac{\partial u}{\partial \nu} + \gamma u = 0$.Comment: 16 page

The heart of a convex body

We investigate some basic properties of the {\it heart} $\heartsuit(\mathcal{K})$ of a convex set $\mathcal{K}.$ It is a subset of $\mathcal{K},$ whose definition is based on mirror reflections of euclidean space, and is a non-local object. The main motivation of our interest for $\heartsuit(\mathcal{K})$ is that this gives an estimate of the location of the hot spot in a convex heat conductor with boundary temperature grounded at zero. Here, we investigate on the relation between $\heartsuit(\mathcal{K})$ and the mirror symmetries of $\mathcal{K};$ we show that $\heartsuit(\mathcal{K})$ contains many (geometrically and phisically) relevant points of $\mathcal{K};$ we prove a simple geometrical lower estimate for the diameter of $\heartsuit(\mathcal{K});$ we also prove an upper estimate for the area of $\heartsuit(\mathcal{K}),$ when $\mathcal{K}$ is a triangle.Comment: 15 pages, 3 figures. appears as "Geometric Properties for Parabolic and Elliptic PDE's", Springer INdAM Series Volume 2, 2013, pp 49-6

Second-order L2L^2-regularity in nonlinear elliptic problems

A second-order regularity theory is developed for solutions to a class of quasilinear elliptic equations in divergence form, including the $p$-Laplace equation, with merely square-integrable right-hand side. Our results amount to the existence and square integrability of the weak derivatives of the nonlinear expression of the gradient under the divergence operator. This provides a nonlinear counterpart of the classical $L^2$-coercivity theory for linear problems, which is missing in the existing literature. Both local and global estimates are established. The latter apply to solutions to either Dirichlet or Neumann boundary value problems. Minimal regularity on the boundary of the domain is required. If the domain is convex, no regularity of its boundary is needed at all

On the regularity up to the boundary for certain nonlinear elliptic systems

We consider a class of nonlinear elliptic systems and we prove regularity up to the boundary for second order derivatives. In the proof we trace carefully the dependence on the various parameters of the problem, in order to establish, in a further work, results for more general systems

The Spherically Symmetric Standard Model with Gravity

Spherical reduction of generic four-dimensional theories is revisited. Three different notions of "spherical symmetry" are defined. The following sectors are investigated: Einstein-Cartan theory, spinors, (non-)abelian gauge fields and scalar fields. In each sector a different formalism seems to be most convenient: the Cartan formulation of gravity works best in the purely gravitational sector, the Einstein formulation is convenient for the Yang-Mills sector and for reducing scalar fields, and the Newman-Penrose formalism seems to be the most transparent one in the fermionic sector. Combining them the spherically reduced Standard Model of particle physics together with the usually omitted gravity part can be presented as a two-dimensional (dilaton gravity) theory.Comment: 58 pages, 2 eps figure

Pathological consequences of VCP mutations on human striated muscle

*These authors have contributed equally to this work. Mutations in the valosin-containing protein (VCP, p97) gene on chromosome 9p13-p12 cause a late-onset form of autosomal dominant inclusion body myopathy associated with Paget disease of the bone and frontotemporal dementia (IBMPFD). We report on the pathological consequences of three heterozygous VCP (R93C, R155H, R155C) mutations on human striated muscle. IBMPFD skeletal muscle pathology is characterized by degenerative changes and filamentous VCP-and ubiquitin-positive cytoplasmic and nuclear protein aggregates. Furthermore, this is the first report demonstrating that mutant VCP leads to a novel form of dilatative cardiomyopathy with inclusion bodies. In contrast to post-mitotic striated muscle cells and neurons of IBMPFD patients, evidence of protein aggregate pathology was not detected in primary IBMPFD myoblasts or in transient and stable transfected cells using wild-type-VCP and R93C-, R155H-, R155C-VCP mutants. Glutathione S-transferase pull-down experiments showed that all three VCP mutations do not affect the binding to Ufd1, Npl4 and ataxin-3. Structural analysis demonstrated that R93 and R155 are both surface-accessible residues located in the centre of cavities that may enable ligand-binding. Mutations at R93 and R155 are predicted to induce changes in the tertiary structure of the VCP protein. The search for putative ligands to the R93 and R155 cavities resulted in the identification of cyclic sugar compounds with high binding scores. The latter findings provide a novel link to VCP carbohydrate interactions in the complex pathology of IBMPFD. Keywords: VCP; p97; myopathy; cardiomyopathy; IBMPFD Abbreviations: GST = glutathione S-transferase; IBMPFD = inclusion body myopathy associated with Paget disease of the bone and frontotemporal dementia; PBS = phosphate-buffered saline; SDS = sodium dodecyl sulphate; VCP = valosin-containing protei

Twenty years of coordination technologies: State-of-the-art and perspectives

Since complexity of inter- and intra-systems interactions is steadily increasing in modern application scenarios (e.g., the IoT), coordination technologies are required to take a crucial step towards maturity. In this paper we look back at the history of the COORDINATION conference in order to shed light on the current status of the coordination technologies there proposed throughout the years, in an attempt to understand success stories, limitations, and possibly reveal the gap between actual technologies, theoretical models, and novel application needs