Ricerc@Sapienza

We consider a magnetostatic problem in a three-dimensional “cylindrical” domain of Koch type. We prove existence and uniqueness results for both the fractal and pre-fractal problems and we investigate the convergence of the pre-fractal solutions to the limit fractal one. We consider the numerical approximation of the pre-fractal problems via FEM and we give a priori error estimates. Some numerical simulations are also shown. Our long-term motivation includes studying problems that appear in quantum physics in fractal domains.

We study radial sign-changing solutions of a class of fully nonlinear elliptic Dirichlet problems in a ball, driven by the extremal Pucci's operators and with a power nonlinear term. We first determine a new critical exponent related to the existence or nonexistence of such solutions. Then we analyze the asymptotic behavior of the radial nodal solutions as the exponents approach the critical values, showing that new concentration phenomena occur. Finally we define a suitable weighted energy for these solutions and compute its limit value.

Numerical and analytical investigations into the aeroelastic response of thin nanocomposite panels are discussed. The mechanical data of the nanocomposite plates with different weight fractions of randomly oriented carbon nanotubes (CNT) are obtained via an extensive experimental campaign based on dynamic mechanical analysis (DMA). Panel flutter induced by high supersonic flows is studied through both linear and nonlinear dynamic analyses aimed to investigate the effect of the CNT weight fraction onto the flutter and post-flutter condition.

A rich variety of resonance scenarios induced in thin elastic plates by periodic axial strains is investigated to describe the morphing capability of flexible, ultra-lightweight panels. The strain-induced excitation is provided by embedded piezoelectric strips actuated by time-varying voltages. The study deals with plates characterized by geometric and mechanical symmetry entailing cylindrical motions so that an ad hoc approximate nonlinear model of elastic beams is adopted to formulate the incremental equation of motion about the nonlinear equilibrium under dead loads.

For any smooth bounded domain (Formula presented.), we consider positive solutions to (Formula presented.)which satisfy the uniform energy bound (Formula presented.)for (Formula presented.). We prove convergence to (Formula presented.) as (Formula presented.) of the (Formula presented.)-norm of any solution. We further deduce quantization of the energy to multiples of (Formula presented.), thus completing the analysis performed in De Marchis et al. (J Fixed Point Theory Appl 19:889–916, 2017).

We prove that the Morse index of any radial solution with m nodal domains of the Lane-Emden problem in the unit ball of R^N, N>2,
is m + N(m ? 1), if the exponent p is sufficiently close to the critical Sobolev exponent.

Semilinear elliptic equations, positive solutions, asymptotic behavior