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We discuss a finite-plasticity model based on the symmetric tensor $P^T P$ instead of the classical plastic strain $P$. Such a model structure arises from assuming that the material behavior is invariant with respect to frame transformations of the intermediate configuration. The resulting variational model is lower-dimensional, symmetric, and based solely on the reference configuration. We discuss the existence of energetic solutions both at the material-point level and for the quasistatic boundary-value problem. These solutions are constructed as limits of time discretizations. Eventually, the linearization of the model for small deformations is ascertained via a rigorous evolutive-$\Gamma$-convergence argument., Comment: 47 pages

A rate-independent model for the quasistatic evolution of a magnetoelastic thin film is advanced and analyzed. Starting from the three-dimensional setting, we present an evolutionary $\Gamma$-convergence argument in order to pass to the limit in one of the material dimensions. By taking into account both conservative and dissipative actions, a nonlinear evolution system of rate-independent type is obtained. The existence of so-called {\it energetic solutions} to such system is proved via approximation.

Symbiotic X-ray binaries form a subclass of low-mass X-ray binary systems consisting of a neutron star accreting material from a red giant donor star via stellar wind or Roche lobe overflow. Only a few confirmed members are currently known; 4U 1700+24 is a good candidate as it is a relatively bright X-ray object, possibly associated with the late-type star V934 Her. We analysed the archive {\it XMM}-Newton and Swift/XRT observations of 4U 1700+24 in order to have a uniform high-energy ($0.3-10$ keV) view of the source. We confirmed the existence of a red-shifted O VIII Ly-$\alpha$ transition (already observed in the 2002 {\it XMM}-Newton data) in the high-resolution spectra collected via the RGS instruments. The red-shift of the line is found in all the analysed observations and, on average, it was estimated to be $\simeq 0.009$. We also observed a modulation of the centroid energy of the line on short time scales (a few days) and discuss the observations in the framework of different scenarios. If the modulation is due to the gravitational red-shift of the neutron star, it might arise from a sudden re-organization of the emitting $X$-ray matter on the scale of a few hundreds of km. Alternatively, we are witnessing a uni-polar jet of matter (with typical velocity of $1000-4000$ km s$^{-1}$) possibly emitted by the neutron star in an almost face-on system. The second possibility seems to be required by the apparent lack of any modulation in the observed $X$-ray light curve. We also note also that the low-resolution spectra (both {\it XMM}-Newton and Swift/XRT in the $0.3-10$ keV band) show the existence of a black body radiation emitted by a region (possibly associated with the neutron star polar cap) with typical size from a few tens to hundreds of meters. The size of this spot-like region reduces as the overall luminosity of 4U 1700+24 decreases., Comment: In press on A&A

We consider a particular class of equations of motion, generalizing to n degrees of freedom the "dissipative spin--orbit problem", commonly studied in Celestial Mechanics. Those equations are formulated in a pseudo-Hamiltonian framework with action-angle coordinates; they contain a quasi-integrable conservative part and friction terms, assumed to be linear and isotropic with respect to the action variables. In such a context, we transfer two methods determining quasi-periodic solutions, which were originally designed to analyze purely Hamiltonian quasi-integrable problems. First, we show how the frequency map analysis can be adapted to this kind of dissipative models. Our approach is based on a key remark: the method can work as usual, by studying the behavior of the angular velocities of the motions as a function of the so called "external frequencies", instead of the actions. Moreover, we explicitly implement the Kolmogorov's normalization algorithm for the dissipative systems considered here. In a previous article, we proved a theoretical result: such a constructing procedure is convergent under the hypotheses usually assumed in KAM theory. In the present work, we show that it can be translated to a code making algebraic manipulations on a computer, so to calculate effectively quasi-periodic solutions on invariant tori. Both the methods are carefully tested, by checking that their predictions are in agreement, in the case of the so called "dissipative forced pendulum". Furthermore, the results obtained by applying our adaptation of the frequency analysis method to the dissipative standard map are compared with some existing ones in the literature.

We investigate a variational theory for magnetoelastic solids under the incompressibility constraint. The state of the system is described by deformation and magnetization. While the former is classically related to the reference configuration, magnetization is defined in the deformed configuration instead. We discuss the existence of energy minimizers without relying on higher-order deformation gradient terms. Then, by introducing a suitable positively $1$-homogeneous dissipation, a quasistatic evolution model is proposed and analyzed within the frame of energetic solvability., Comment: 11 pages

We present a stability result for a wide class doubly nonlinear equations, featuring general maximal monotone operators, and (possibly) nonconvex and nonsmooth energy functionals. The limit analysis resides on the reformulation of the differential evolution as a scalar energy-conservation equation with the aid of the so-called Fitzpatrick theory for the representation of monotone operators. In particular, our result applies to the vanishing viscosity approximation of rate-independent systems.

We provide a rigorous justification of the classical linearization approach in plasticity. By taking the small-deformations limit, we prove via \Gamma-convergence for rate-independent processes that energetic solutions of the quasi-static finite-strain elastoplasticity system converge to the unique strong solution of linearized elastoplasticity., Comment: To appear on J. Eur. Math. Soc. (JEMS)

We consider a general formulation of gradient flow evolution for problems whose natural framework is the one of metric spaces. The applications we deal with are concerned with the evolution of {\it capacitary measures} with respect to the $\gamma$-convergence dissipation distance and with the evolution of domains in spectral optimization problems., Comment: 24 pages, 0 figures

We prove a conjecture by De Giorgi on the elliptic regularization of semilinear wave equations in the finite-time case., Comment: 16 pages, to appear on Math. Models. Meth. Appl. Sci

We develop the long-time analysis for gradient flow equations in metric spaces. In particular, we consider two notions of solutions for metric gradient flows, namely energy and generalized solutions. While the former concept coincides with the notion of curves of maximal slope, we introduce the latter to include limits of time-incremental approximations constructed via the Minimizing Movements approach. For both notions of solutions we prove the existence of the global attractor. Since the evolutionary problems we consider may lack uniqueness, we rely on the theory of generalized semiflows introduced by J.M. Ball. The notions of generalized and energy solutions are quite flexible and can be used to address gradient flows in a variety of contexts, ranging from Banach spaces to Wasserstein spaces of probability measures. We present applications of our abstract results by proving the existence of the global attractor for the energy solutions both of abstract doubly nonlinear evolution equations in reflexive Banach spaces, and of a class of evolution equations in Wasserstein spaces, as well as for the generalized solutions of some phase-change evolutions driven by mean curvature.

We present a variational principle governing the quasistatic evolution of a linearized elastoplastic material. In case of linear hardening, the novel characterization allows to recover and partly extend some known results and proves itself to be especially well-suited for discussing general approximation and convergence issues. In particular, the variational principle is exploited in order to prove in a novel setting the convergence of time and space-time discretizations as well as to provide some possible a posteriori error control., Comment: 29 pages, 1 figure

This note addresses a three-dimensional model for isothermal stress-induced transformation in shape-memory polycrystalline materials. We treat the problem within the framework of the energetic formulation of rate-independent processes and investigate existence and continuous dependence issues at both the constitutive relation and quasi-static evolution level. Moreover, we focus on time and space approximation as well as on regularization and parameter asymptotics., Comment: 33 pages, 3 figures

This paper addresses the long-time behavior of gradient flows of non convex functionals in Hilbert spaces. Exploiting the notion of generalized semiflows by J. M. Ball, we provide some sufficient conditions for the existence of a global attractor. The abstract results are applied to various classes of non convex evolution problems. In particular, we discuss the long-time behavior of solutions of quasi-stationary phase field models and prove the existence of a global attractor., Comment: 46 pages

We address the self-similar solvability of a singular parabolic problem and show that solutions to parabolic fully nonlinear equations are not expected to be $C^{2,1}$., Comment: 15 pages, 5 figures

This paper addresses a doubly nonlinear parabolic inclusion of the form $A(u_t)+B(u)\ni f$. Existence of a solution is proved under suitable monotonicity, coercivity, and structure assumptions on the operators $A$ and $B$, which in particular are both supposed to be subdifferentials of functionals on $L^2(\Omega)$. Moreover, under additional hypotheses on $B$, uniqueness of the solution is proved. Finally, a characterization of $\omega$-limit sets of solutions is given and we investigate the convergence of trajectories to limit points.