The paper studies continuity properties of the functional in the Moser-Trudinger inequality on the Sobolev space W^1,N of domains (not necessarily bounded) in ℝ^N . Given that the functional is not weakly continuous, one may describe its asymptotic properties on bounded sequences in W^1,N in terms of concentration compactness and, more specifically, in the terms of a profile decomposition. While the well-known profile decomposition in W^1,p(ℝ^N) expresses the concentration as a sum of elementary concentrations (“bubbles”) of the form t_k^N-p/pw(t(x-y_k)) , elementary concentrations in the case p=N have the form s_k^N-1/Nw(|x-y_k|^1/s_k) , s_k→∞ , with always radial profiles w. We also prove that nonlinear functional in the Moser-Trudinger inequality fails to be weakly continuous only on exceptional sequences.

In the context of thermo-elasticity we consider initial boundary value problems governed by parabolic and hyperbolic heat propagations. In particular, we describe the evolution of the temperature and displacement fields in a one dimensional string attached to a rigid substrate through an adhesive layer. This adhesive interaction is characterized by a nonlinear term describing the adhesion force exhibiting discontinuities when a critical value of the displacement is reached, in the limit of parabolic heat propagation. We study the well-posedness of the problem under Neumann boundary conditions in the two different regimes of heat propagation and investigate the long time dynamics.

Design of piles under lateral loads requires accurate estimation of the pile deflections. The Winkler beam model is the most popular mechanical model for piles, employed by both researchers and engineers. However, it assumes a constant soil stiffness, which is unrealistic since experimental data, referred to homogeneous soils, prove a nonlinear trend of the soil stiffness with depth. This paper discusses the limits of existing pile models and proposes an unprecedented analytical model for piles under lateral loads embedded in soils with stiffness that does not linearly depend on depth. The analytical solution of the governing Ordinary Differential Equation is derived and represented in explicit closed-form in terms of generalized hypergeometric functions. In addition, the paper delivers parametrized solutions for typical load conditions where a horizontal force and bending moment are applied at the pile's top. The paper examines a pile response in three cases assuming as soil stiffness the constant, linear and cubic fitting of experimental estimates of the subgrade stiffness of an actual clay deposit. No negligible difference in displacement, moment and shear were detected. Still, the proposed nonlinear model leads to the most conservative estimates of the displacement. Peculiar trends characterize the results in shear and bending moment responses compared to the constant and linear soil profiles.

In this paper we extend to the case of a generic dimension N > 2 some notions introduced in a previous article to establish some new variational methods. Such notions relay on the concept of higher order barycenters of a positive measure and allow to give some symmetry conditions which in two dimensions lead to the conclusions that the measures which satisfy them in a suitably optimal way are uniformly concentrated on the vertices of a regular polygon. Here, investigating the extension of such properties to the case of a general dimension, we mainly focus on the geometrical and algebraic aspects, including some connections with Platonic Solids regardless any further application to variational methods. (c) 2022 Elsevier Masson SAS. All rights reserved.

We discuss some attempts to apply the classical variational methods, in the spirit of Ambrosetti–Rabinowitz, to parametrized elliptic problems defined on unbounded domains. We construct some min-max classes and establish some estimates which allow, for instance, to conclude that, while for a generic coefficient there is always a solution for all small values of the parameter, in the case of a coefficient with a suitable exponential decay and in dimension $N=2$ the set of the values of the parameter for which the problem has a nontrivial solution is unbounded.

By using the preliminary results given in a previous divulgative note, we present here a concise and self-contained introduction to the construction of the real field as the unique, up to increasing isomorphism, Dedekind complete totally ordered field. Moreover, we also show the equivalence between the Dedekind completeness property on totally ordered fields and some meaningful well-known notions present in the literature, such as the Cauchy completeness on totally ordered Archimedean fields. This characterization result allows us to correctly encode the Dedekind completeness for totally ordered fields in the general abstract setting of metric spaces. We believe that the essential parts of the paper can be easily accessed by anyone with some experience in abstract mathematical thinking. The paper completes the lecture given by the second author during the International Workshop on New Horizons in Teaching Science in Messina on June 2018.

These notes reflect partially the contents of a lecture given by the second author during the International Workshop on New Horizons in Teaching Science in Messina on June 2018. The intention of this lecture was to present a concise and self–contained introduction to the construction of the real field as the unique, up to increasing isomorphism, Dedekind complete totally ordered field.

We study a prototypical system describing instability effects due to geometric constraints in the framework of nonlinear elasticity. By considering the equilibrium configurations of an elastic ring constrained inside a rigid circle with smaller radius, we analytically determine different possible shapes, reproducing well-known physical phenomena. As we show, both single- (with different complexity) and multi-blister configurations can be observed, but the lowest energy always corresponds to single-blister solutions. Important physical insight is attained through an analogy between the elastica and the dynamics of a nonlinear pendulum. A complete geometric characterization is attained, proving symmetry and other relevant properties. The effectiveness of the model is tested against a simple experiment by considering a thin polymer strip constrained in a rigid cylinder.

The paper studies the initial boundary value problem related to the dynamic evolution of an elastic beam interacting with a substrate through an elastic-breakable forcing term. This discontinuous interaction is aimed to model the phenomenon of attachment-detachment of the beam occurring in adhesion phenomena. We prove existence of solutions in energy space and exhibit various counterexamples to uniqueness. Furthermore we characterize some relevant features of the solutions, ruling the main effects of the nonlinearity due to the elastic-breakable term on the dynamical evolution, by proving the linearization property according to Gérard (J Funct Anal 141(1):60–98, 1996) and an asymptotic result pertaining the long time behavior.

We study some variational problems related to the controllability of the transition between two quantum states of the driven quantum harmonic oscillator. We focus on the existence of minimum points of the total energy functional which is obtained by adding to the cost of the control a term penalizing the distance from the maximum transition probability between given energy states.

This paper deals with some recent results for scalar field equations in the case of a potential which converges from above to a limit at infinity. In particular, we shall collect some of the results of the authors which have contributed to partially answer a conjecture of Wei and Yan and we shall announce some results in a forthcoming paper to the aim of giving a short survey which collects the main ideas based on the use of symmetry assumptions and of symmetry methods introduced for the above problem.

Growth-induced pattern formation in tubular tissues is intimately correlated to normal physiological functions. Moreover, either the microstructure or certain diseases can give rise to material inhomogeneity, which can lead to a change of shape in the tissue. Therefore, it is of fundamental importance to understand surface instabilities and pattern transitions of graded tubular tissues. In this paper we perform such analysis by the use of a mechanical model of a graded tube which grows with a fixed outer boundary by focusing on a plane-strain problem within the framework of nonlinear elasticity. A theoretical model is established to determine the uniform growth state, the critical growth factor, and the critical wavenumber for a general material model and for a general material gradient. For a case study, the material is specified by the neo-Hookean model, and the shear modulus is assumed to decay linearly or exponentially from the inner surface. Then, a parametric study is carried out to unravel the effects of material and geometrical parameters on the bifurcation threshold and the associated wrinkled pattern. In addition, a finite element model, which is validated by the theoretical one, is developed to trace the post-buckling evolution. It is found that wrinkled pattern will evolve into an arch mode and then into a creasing mode if the modulus decays linearly. However, the typical creasing mode may give way to a period-doubling mode when applying an exponentially decaying modulus, and there is a co-existence of the creasing mode and the wrinkling mode. As a result, different modulus gradients can generate diverse pattern formations. The obtained results are useful to supply insight into the effects of material inhomogeneity and different modulus gradients on surface instabilities and morphology evolutions in graded tubular tissues.

In the present paper, we show how to define suitable subgroups of the orthogonal group $${O}(d-m)$$ related to the unbounded part of a strip-like domain $$\omega \times {\mathbb {R}}^{d-m}$$ with $$d\ge m+2$$ , in order to get “mutually disjoint” nontrivial subspaces of partially symmetric functions of $$H^1_0(\omega \times {\mathbb {R}}^{d-m})$$ which are compactly embedded in the associated Lebesgue spaces. As an application of the introduced geometrical structure, we prove (existence and) multiplicity results for semilinear elliptic problems set in a strip-like domain, in the presence of a nonlinearity which either satisfies the classical Ambrosetti–Rabinowitz condition or has a sublinear growth at infinity. The main theorems of this paper may be seen as an extension of existence and multiplicity results, already appeared in the literature, for nonlinear problems set in the entire space $${\mathbb {R}}^d$$ , as for instance, the ones due to Bartsch and Willem. The techniques used here are new.

The paper studies existence of solutions for the nonlinear Schrödinger equation(0.1)−(∇+iA(x))2u+V(x)u=f(|u|)u with a general bounded external magnetic field. In particular, no lattice periodicity of the magnetic field or presence of external electric field is required. Solutions are obtained by means of a general structural statement about bounded sequences in the magnetic Sobolev space.

The paper deals with the harvesting model introduced by one of the authors in a joint paper with A. Bressan and W. Shen. A cost term which plays an essential role in the existence proof is removed here leading to some new phenomena. Since a solution not always exists in this case, we propose a relaxed version of the problem which is always solvable, has a geometrical interpretation in the functions space and whose unique solution, under suitable conditions, solves also the original problem. Finally, the notion of strategy limit of a sequence of smooth measures is introduced as a long-term control to fish populations.

The paper studies existence of solutions for the nonlinear Schrdinger equation (0.1) − (∇+ iA(x))u+ V (x)u = f(|u|)u with a general bounded external magnetic field. In particular, no lattice periodicity of the magnetic field or presence of external electric field is required. Solutions are obtained by means of a general structural statement about bounded sequences in the magnetic Sobolev space.

For many known non-compact embeddings of two Banach spaces $E\hookrightarrow F$, every bounded sequence in $E$ has a subsequence that takes form of a profile decomposition - a sum of clearly structured terms with asymptotically disjoint supports plus a remainder that vanishes in the norm of $F$. In this note we construct a profile decomposition for arbitrary sequences in the Sobolev space $H^{1,2}(M)$ of a compact Riemannian manifold, relative to the embedding of $H^{1,2}(M)$ into $L^{2^*}(M)$, generalizing the well-known profile decomposition of Struwe ([Proposition 2.1]{Struwe}) to the case of arbitrary bounded sequences.

The paper is related to a conjecture by Pegon, Santambrogio and Xia concerning the dimension of the boundary of some sets which we are calling "irrigation balls". We propose a notion of sub-balls and sub-spheres of prescribed radius and we prove that, generically, the only possible Minkowski dimension of sub-spheres is the one expected in the conjecture. At the same time, beside the scale transition properties and the dimension estimates on some significant sets, we propose a third approach to study the fractal regularity which relies on lower oscillation estimates on the landscape function, which turns out to behave as a Weierstrass-type function.

The aim of this paper is to provide a useful tool for a better understanding of the approach proposed in [11] to extend to the setting of metric spaces prole decomposition theorems. To this aim we shall deal with a less general context which has the advantage of making the analogies with the linear case more evident.

This paper contains a survey on one of the mathematical approaches used to solve a fractional differential equation whose solution gives the free dynamic response of viscoelastic single degree of freedom systems (viscosity is actually modelled by a fractional displacement derivative instead of first order one). The paper shall deal with Caputo’s fractional derivative since its Laplace Transform (on which the resolution method is based) only depends by lower integer order (and therefore measurable and physically meaningful) derivatives given as initial conditions. The paper provides a deep mathematical analysis of the properties of the solution expressed in terms of the mechanical parameters meaning. Additionally, important physical implications are reported exhibiting a richer dynamic behavior if compared to the standard damping case (velocity linear dependence). Some important consequences in the use of Caputo’s fractional derivative are reported, and some limitations to possible viscous parameters values are obtained. Finally, it is shown that free response of fractional derivative equation solution is mathematically equivalent to a suitably forced solution of the integer model.