We prove that every entire solution with quadratic growth, lying in a suitable cone, to the 2-Monge-Ampère equation on $\mathbb{R}^3$ is a quadratic polynomial. The proof proceeds by first establishing a concavity inequality, and then deriving a Pogorelov-type interior $C^2$ estimate.
We study the uniqueness of non-negative solutions of the equation \begin{align*} \partial_t\left(|u|^{p-2}u\right)\,=\, \operatorname{div}(|\nabla u|^{p-2}\nabla u). \end{align*} Basic estimates are derived with the Galerkin Method.
We prove an abstract linking theorem that can be used to show existence of solutions to various types of variational elliptic equations, including Schrödinger--Poisson--Slater type equations.
We study positive singular solutions of the Loewner-Nirenberg problem on conical domains and establish the existence of solutions that admit prescribed asymptotic expansions near vertices, valid to arbitrarily high order of approximation.
We perform a mass constrained phase-field approximation for a model that describes the epitaxial growth of a two-dimensional thin film on a substrate in the context of linearised elasticity. The approximated model encodes a variable on the free surface of the film, that physically is interpreted as an adatom density.
We construct non-trivial steady solutions in $H^{-1}$ for the 2D Navier-Stokes equations on the torus. In particular, the solutions are not square integrable, so that we have to redefine the notion of solutions.
We present a general framework for constructing examples on Lavrentiev energy gap for nonlocal problems and apply it to several nonlocal and mixed models of double-phase type.
In this work we prove a Strichartz estimate for the Schrödinger equation in the quasiperiodic setting. We also show a lower bound on the number of resonant frequency interactions in this situation.
Public Archaeology is a young discipline, we all know that. It’s even younger in Italy, where public archaeology has not even reached ‘adulthood’. Cited for the first time by Armando De Guio in 2000 (De Guio and Bressan 2000), it was only a decade later that Public Archaeology has started to become ‘a thing’, thanks to some pioneering experiences at the University of Florence (Bonacchi 2009; Vannini 2011), and especially after a national conference in 2012 (in Florence: see Zuanni 2013 for a summary). Italian archaeologists’ first reaction was to overlap the new discipline with the experiences already in place, which in Italy were under the category of ‘valorizzazione’ (enhancement). They were not exactly the same: while Public Archaeology is characterised by a reflection on the objectives of the research from the very start, a focus on having a reliable methodology, and a strong element linked to evaluation, ‘enhancement’ experiences – while often valuable and successful – lacked the same structure and reliability. This is probably due to an underestimation of these practices as a scientific topic, thus deserving the same structure required for any other type of research. Often this resulted in a mere description of the activities carried out, with a generic objective like ‘increasing the knowledge of archaeology in the public sphere’ without really evaluating if the activities worked or not. Public Archaeology became a sort of a trendy subject, outdating the term ‘valorizzazione’, at least in most of the university milieu, and creating confusion on the subject and the methodology. This sometimes has led to a sort of ‘hangover’ effect, similar to what happens with summer songs: they sound fun when you first hear them, but after months you just want to move on! Few doctoral theses awarded in Archaeology have been devoted to topics related to Public Archaeology up to the present date and the risk is that after this ‘hangover’ the subject will be penalised in comparison to others.
In this note we complete the study made in a previous paper on a Kirchhoff type equation with a Berestycki-Lions nonlinearity. We also correct Theorem 0.6 inside.
In this paper we investigate some convergence and divergence properties of the logarithmic means of quadratical partial sums of double Fourier series of functions in the measure and in the $L$ Lebesgue norm.
Motivated by a recent result which identifies in the special setting of the 2-adic group the Besov space $\dot{B}^{1,\infty}_{1}(\mathbb{Z}_2)$ with $BV(\mathbb{Z}_2)$, the space of function of bounded variation, we study in this article some functional relationships between Besov spaces.
In this article we study an elliptic problem with degenerate coercivity. We will show that the presence of some lower order terms has a regularizing effect on the solutions.
I prove that the time derivative for the solution of the obstacle problem related to the Evolutionary p-Laplace Equation exists in Sobolev's sense, provided that the given obstacle is smooth enough. We keep p > 2.
Given $p$ points in a bounded domain of $\R^d$, with $d=2,3$, we show the existence of solutions of the $L^2$-critical focusing nonlinear Schrödinger equation blowing up exactly in these points.
We prove global comparison results for the $p$-Laplacian on a $p$-parabolic manifold. These involve both real-valued and vector-valued maps with finite $p$-energy.