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We study a shape optimization problem for the first eigenvalue of an elliptic operator in divergence form, with non constant coefficients, over a fixed domain $\Omega$. Dirichlet conditions are imposed along $\partial \Omega$ and, in addition, along a set $\Sigma$ of prescribed length ($1$-dimensional Hausdorff measure). We look for the best shape and position for the supplementary Dirichlet region $\Sigma$ in order to maximize the first eigenvalue. The limit distribution of the optimal sets, as their prescribed length tends to infinity, is characterized via $\Gamma$-convergence of suitable functionals defined over varifolds: the use of varifolds, as opposed to probability measures, allows one to keep track of the local orientation of the optimal sets (which comply with the anisotropy of the problem), and not just of their limit distribution., Comment: 23 pages, 2 figures

We investigate the existence of ground states for the subcritical NLS energy on metric graphs. In particular, we find out a topological assumption that guarantees the nonexistence of ground states, and give an example in which the assumption is not fulfilled and ground states actually exist. In order to obtain the result, we introduce a new rearrangement technique, adapted to the graph where it applies. Owing to such a technique, the energy level of the rearranged function is improved by conveniently mixing the symmetric and monotone rearrangement procedures., Comment: 24 pages, 4 figures

Analysis of measured X-ray diffraction (XRD) data from heterostructures with fitting analysis is discussed, for which computer program was written. Lattice constant and Poisson's ratio of a multi-compound layer is calculated from Vegard's law. Layers are strained in in-plane direction causing additional strain in out-of-plane direction and may be only partially strained. For crystals, wavelength-dependent electric susceptibility is represented as Fourier series, components of which are calculated from atomic scattering factors, H\"onl anomalous dispersion correction terms and Debye-Waller factor. XRD intensity from heterostructures as a function of angle is calculated with dynamical diffraction theory using Takagi-Taupin differential equation solution in form of recursive equations. XRD measurements are analyzed by fitting a simulated XRD curve to the measured XRD curve. Inverse problem related to curve fitting is solved with differential evolution where principal component analysis is used to reduce interparameter dependencies. For measurement analysis a software was written with Java/Matlab, user interface (UI) of which was designed by taking into account feedback from users. With UI, it is really easy to build a layer model, parameters of which are obtained by fitting a simulation to measurement. Software is also able to use GNU Octave and therefore works without requiring any commercial components. Operability of simulation code of XRD curves was tested by comparing the curves simulated by it with curves simulated by 2 other software using single/multilayer InGaAs, GaAs and GaAsN structures. Software was tested by analyzing a measurement series having been published previously. Software was observed to produce results similar to other software and the fitting algorithm was observed to work efficiently., Comment: 48 pages, in Finnish. Abstract shortened and Finnish abstract missing due to abstract length limit

GaAsPN layers with a thickness of 30nm were grown on GaP substrates with metalorganic vapor phase epitaxy to study the feasibility of a single X-ray diffraction (XRD) measurement for full composition determination of quaternary layer material. The method is based on the peak intensity of a quasi-forbidden (002) reflection which is shown to vary with changing arsenic content for GaAsPN. The method works for thin films with a wide range of arsenic contents and shows a clear variation in the reflection intensity as a function of changing layer composition. The obtained thicknesses and compositions of the grown layers are compared with accurate reference values obtained by Rutherford backscattering spectroscopy combined with nuclear reaction analysis measurements. Based on the comparison, the error in the XRD defined material composition becomes larger with increasing nitrogen content and layer thickness. This suggests that the dominating error source is the deteriorated crystal quality due to the nonsubstitutional incorporation of nitrogen into the crystal lattice and strain relaxation. The results reveal that the method overestimates the arsenic and nitrogen content within error margins of about 0.12 and about 0.025, respectively.

We prove the nonexistence of ground states for NLS on bridge-like graphs, i.e. graphs with two halflines and four vertices, of which two at infinity, with Kirchhoff matching conditions. By ground state we mean any minimizer of the energy functional among all functions with the same mass., Comment: 13 pages, 6 figures, Contribution to the proceedings of the Workshop "Mathematical Technology of Networks", ZiF Bielefeld, December 4-7, 2013

Developing an original idea of De Giorgi, we introduce a new and purely variational approach to the Cauchy Problem for a wide class of defocusing hyperbolic equations. The main novel feature is that the solutions are obtained as limits of functions that minimize suitable functionals in space-time (where the initial data of the Cauchy Problem serve as prescribed boundary conditions). This opens up the way to new connections between the hyperbolic world and that of the Calculus of Variations. Also dissipative equations can be treated. Finally, we discuss several examples of equations that fit in this framework, including nonlocal equations, in particular equations with the fractional Laplacian.

In this chapter some results related to Shunt Active Filters (SAFs) and obtained by the authors and some coauthors are reported. SAFs are complex power electronics equipments adopted to compensate for cur-rent harmonic pollution in electric mains, due to nonlinear loads. By using a proper "floating" capacitor as energy reservoir, the SAF purpose is to inject in the line grid currents canceling the polluting har-monics. Control algorithms play a key role for such devices and, in general, in many power electronics applications. Moreover, systems theory is crucial, since it is the mathematical tool that enables a deep understanding of the involved dynamics of such systems, allowing a correct dimensioning, beside an effective control. As a matter of facts, current injection objective can be straightforwardly formulated as an output tracking control problem. In this fashion, the structural and insidious marginally-stable internal/zero dynamics of SAFs can be immediately highlighted and characterized in terms of sizing and control issues. For what concerns the control design strictly, time-scale separation among output and internal dynamics can be effectively exploited to split the control design in different stages that can be later aggregated, by using singular perturbation analysis. In addition, for robust asymptotic output tracking the Internal Model Principle is adopted., Comment: Paper presented at the AUTOMATICA_IT 2011 conference, Pisa, Italy, September 2011