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ANDREA GAVIOLI

OSPITE CON ACCESSO AL SERVIZIO VPN
Dipartimento di Scienze Fisiche, Informatiche e Matematiche sede ex-Matematica

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Pubblicazioni

2020 - On a mathematical model for a damped and driven double-well Bose–Einstein condensate [Articolo su rivista]
Gavioli, A.; Sacchetti, A.
abstract

In this paper we consider a two-mode dynamical system as a model for a driven one-dimensional damped Bose–Einstein condensate in a double-well trapping potential. In the case of a constant external driving force the existence and stability of stationary solutions are discussed in relation to the values of the physical parameters. In the case of a time-dependent periodic external driving force the existence of limit cycles is proved, and the amplitude of these limit cycles exhibits a jump phenomenon for critical values of the physical parameters.

2017 - Heteroclinic connections for a double-well potential with an asymptotically periodic coefficient [Articolo su rivista]
Gavioli, Andrea
abstract

We prove the existence of monotone heteroclinic solutions to a scalar equation of the kind u″=a(t)V′(u) under the following assumptions: V∈C2(R) is a non-negative double well potential which admits just one critical point between the two wells, a(t)a(t) is measurable, asymptotically periodic and such that inf_a>0, sup_a<+∞. In particular, we improve earlier results in the so called asymptotically autonomous case, when the periodic part of a, say \alpha, is constant, i.e. a(t) converges to a positive value l as |t|→+∞. Furthermore, whenever \alpha fulfils a suitable non-degeneracy condition, the solutions are shown to be infinitely many.

2016 - Positive homoclinic solutions to some Schrodinger type equations [Articolo su rivista]
Gavioli, Andrea; Sanchez, Luis
abstract

By means of variational methods we prove the existence of a positive, homoclinic solution to an equation of the kind u''=au-bu^p, where p>1, and both coefficients a(x), b(x) are positive and asymptotically constant. Our main result requires a control from above on the ratios .between the supremum of a(x) and its limit at infinity and between the limit at infinity of b(x) and its infimum.

2015 - A variational property of critical speed to travelling waves in the presence of nonlinear diffusion [Articolo su rivista]
Gavioli, Andrea; Sanchez, Luis
abstract

We consider a reaction-diffusion equation, where the diffusion is governed by a p-Laplacian and the reaction term f is positive on ]0,1[ and vanishes elsewhere. Under additional mild conditions on f we show that the minimal speed for the corresponding travelling waves can be computed via a suitable constrained minimum problem.

2014 - Analytical basis for determining slope lines in grid digital elevation models [Articolo su rivista]
Orlandini, Stefano; Moretti, Giovanni; Gavioli, Andrea
abstract

[1] An analytical basis for the determination of slope lines in grid digital elevation models is provided by using the D8-LTD method (eight slope directions, least transverse deviation). The D8-LTD method’s capability to predict consistently exact slope lines as the grid cell size goes to zero is shown analytically by applying mathematical analysis methods. The use of cumulative, least transverse deviations is found to be the key factor allowing for globally unbiased approximations of slope lines. The D8-LTD method’s properties are also demonstrated numerically by using digital elevation models of a synthetic sloping surface obtained from the Himmelblau function. It is shown that slope lines obtained from the D8-LTD method can approximate the exact slope lines as close as desired by selecting a grid cell size that is small enough. In contrast, the standard D8 method is found to produce significantly biased results even when small grid cells are used. The D8-LTD method outperforms the D8 method over a wide range of grid cell sizes (up to 20 m in this application), beyond which grid cell size becomes too large to validly represent the underlying sloping surface. It is therefore concluded that the D8-LTD method should be used in preference to the standard D8 method in order to obtain slope lines that are only limited in reliability by the detail of topographic data, and not by the accuracy of the slope direction method applied.

2013 - A class of singular first order differential equations with applications in reaction-diffusion [Articolo su rivista]
Enguiça, Ricardo; Gavioli, Andrea; Sanchez, Luis
abstract

We study positive solutions y(u) to the first order differential equation y'=q(cy^{1/p}-f(u)) where c&gt;0 is a parameter, p&gt;1 and q&gt;1 are conjugate numbers and f is a continuous function on [0,1] such that f(0)=0=f(1). We shall be particularly concerned with solutions y(u) such that y(0)=0=y(1). Our motivation lies in the fact that this problem provides a model for the existence of travelling wave solutions for analogues of the FKPP equation in one spacial dimension, where diffusion is represented by the p-Laplacian operator. We obtain a theory of admissible velocities and some other features that generalize classical and recent results, established for p=2.

2011 - Heteroclinic solutions to asymptotically autonomous equations via continuation methods [Articolo su rivista]
Gavioli, Andrea
abstract

By means of a continuation argument, we prove the existence of at least one increasing heteroclinic solution to a scalar equation of the kind x''=a(t)V'(x), where V is a non-negative double well potential, and a(t) is a positive, measurable coefficient, which is definitively monotone with respect to |t|, converges to a positive limit l as |t| diverges and fulfils one of the two following assumptions: a(t) never goes below l, or a(t)-l converges to 0, as |t| diverges, more slowly than a suitable exponential term.

2011 - Monotone heteroclinic solutions to non-autonomous equations via phase plane analysis [Articolo su rivista]
Gavioli, Andrea
abstract

We study the existence of at least one increasing heteroclinic solution to a scalar equation of the kind x''=a(t)V'(x), where V is a non-negative double well potential, and a(t) is a positive, measurable coefficient. We first provide with a complete answer in the definitively autonomous case, when a(t) takes a constant value l outside a bounded interval. Then we consider the case in which a(t) is definitively monotone, converges from above, as t diverges to the left and to the right, to two positive limits, and never goes below the minimum between them. Furthermore, the convergence to the maximum between them is supposed to be not too fast (slower than a suitable exponential term).

2010 - Solutions of second-order and fourth-order ODE's on the half-line [Articolo su rivista]
R., Enguiça; Gavioli, Andrea; L., Sanchez
abstract

We start by studying the existence of positive solutions (on the positive half-line) for the differential equation u''(x)=a(x)u-g(u), under the condition u'(0)=0, and that u vanishes at infinity. The coefficient a(x) is positive, g satisfies suitable growth hypotheses or, in alternative, is bounded. Then we deal with the analogous fourth order problem, where the left-hand side of the equation is replaced by -u''''+cu'' (c>0), g(u)/u is a power of |u|, and the further condition u'''(0)=0 is required.

2009 - Heteroclinics for non-autonomous second-order differential equations [Articolo su rivista]
Gavioli, Andrea; L., Sanchez
abstract

We investigate new conditions for the existence of heteroclinic solutions of a non-autonomous equation of the form u''=a(t)f(u), where a(t) is a bounded, positive function, f(-1)=f(1)=0, and f=F', where F is a C^1, non-negative function such that F(-1)=F(1)=0. We are mainly interested in the case where a(t) approaches its positive limit from above, as |t| diverges, but we allow also the "asymptotically asymmetric" case, where the difference between the two limits (at minus infinity and plus infinity) is a sufficiently small positive number. Variational methods are used in the proof.

2009 - On the existence of heteroclinic trajectories for asymptotically autonomous equations [Articolo su rivista]
Gavioli, Andrea
abstract

By means of a minimax argument, we prove the existence of at least one heteroclinic solution to a scalar equation of the kind x''=a(t)V'(x), where V is a double well potential, 0<l<=a(t)<=L, a(t) converges to l as |t| diverges and the ratio L/l is suitably bounded from above.

2008 - On a class of bounded trajectories for some non-autonomous systems [Articolo su rivista]
Gavioli, Andrea; L., Sanchez
abstract

We consider on the positive half-line an equation of the kindx''+cx'=a(t)V'(x) and prove, by variational arguments, the existence of a solution which fulfils the boundary conditions x(0)=0, x(+\infty)=1. The constant c is non-negative and a(t) belongs to a certain class of positive functions. The existence of such a solution in the case c=0 means that the system behaves in a significantly different way from its autonomous counterpart.

2007 - On Bounded Trajectories for Some Non-Autonomous Systems [Relazione in Atti di Convegno]
Gavioli, Andrea; L., Sanchez
abstract

We recall conditions for the existence of heteroclinics connecting the points -1 and 1 for a non-autonomous equation of the form u''=a(t)f(u), where a(t) is a bounded positive function such that f(-1)=f(1)=0. In addition, we consider the existence of a solution to the boundary value problem in the half line

2007 - Trends in Differential Equations and Dynamical Systems [Esposizione]
Gavioli, Andrea; Malaguti, Luisa; Villarini, Massimo
abstract

The workshop took place at Modena, from November 29th to 30th.The main speakers were J. Andres, from Palachy University (Olomouc, SK), P. K. Maini, from the University of Oxford (UK), V. Obukhovskii, from Voronezh State University (Russia), and other speakers from Italy (R. Johnson, F. Papalini, M. Tarallo). Furthermore, several talks were given by young researchers. The topics of the meeting covered many different areas in the field of differential equations and related problems. In particular, here are some of the exposed subjects: front-propagation in reaction-diffusion equations, which often arise from biological models, Sturm-Liouville operators, perturbation theory for Hamiltonian systems, impulsive control systems, boundary value problems and the theory of bound sets, delay equations, differential inclusions.

2005 - Trends in Differential Equations and Dynamical Systems [Esposizione]
Gavioli, Andrea; Malaguti, Luisa; Villarini, Massimo
abstract

The workshop took place at Reggio Emilia, from September 29th to 30th. The main speakers were S. Kamin from Tel-Aviv University, Tel-Aviv (Israel) and L. Sanchez from Lisbon University (Lisbon, Portugal) and other speakers from Italy (A. Agrachev and S. Terracini). Furthermore, several talks were given by young researchers. The topics of the meeting covered many different areas in the field of differential equations and control problems.

2004 - Existence of periodic orbits for vector fields via Fuller index and the averaging method [Articolo su rivista]
P., Benevieri; Gavioli, Andrea; Villarini, Massimo
abstract

We prove a generalization of a theorem proved by Seifert and Fuller concerning the existence of periodic orbits of vector fields via the averaging method. Also we show applications of these results to Kepler motion and to geodesic flows on spheres.

2002 - Hybrid stabilization of planar linear systems with one-dimensional outputs [Articolo su rivista]
Benassi, Carlo 6/8/1962; Gavioli, Andrea
abstract

We consider a linear control system x'=Ax+Bu with output y=Cx, where x is two-dimensional, u,y are one-dimensional, and give necessary and sufficient conditions in order that it can be stabilized by a hybrid, linear feedback, where the action of the "switch" just depends on the sign of y. We also show, on these conditions, that the use of two control functions is enough for getting the goal.

2000 - Approximation from the Exterior of Caratheodory Multifunctions [Articolo su rivista]
Benassi, Carlo 6/8/1962; Gavioli, Andrea
abstract

We approximate a globally measurable multifunction F(t,x) which takes compact values in a euclidean space by means of a decreasing sequence of globally measurable multifunctions F_n(t,x) which are locally lipschitzian with respect to x, in the following cases: F(t,\cdot) is upper semicontinuous and takes connected values, or F(t,\cdot) is continuous.

2000 - Viable solutions of differential inclusions with memory in Banach spaces [Articolo su rivista]
Gavioli, Andrea; Malaguti, Luisa
abstract

In this paper we study functional differential inclusions with memory and state constraints. We assume the state space to be a separable Banach space and prove existence results for an upper semicontinuous orientor field; we consider both the case of a globally measurable orientor field and the case of a Caratheodory one.

1999 - On the Solution Set of the Nonconvex Sweeping Process [Articolo su rivista]
Gavioli, Andrea
abstract

We prove that the solutions of a sweeping process make up an R_{\d}-set under the following assumptions: the moving set C(t) has a lipschitzian retraction and, in the neighbourhood of each point x of its boundary, it can be seen as the epigraph of alipschitzian function, in such a way that the diameter of the neighbourhood and the related Lipschitz constant do not depend on x and t. An application to the existence of periodic solutions is given.

1998 - A Viability Result in the Upper Semicontinuous Case [Articolo su rivista]
Gavioli, Andrea
abstract

We prove the existence of solutions of a differentialinclusion u'\in F(t,u) in a separable Banach space X with a moving constraint D(t). F is globally measurable, weakly upper semicontinuous with respect to u and takes convex, weakly compact values. D is upper semicontinuous from the left, and, for every r>0, the sets D(t)\cap rB are compact. F and D fulfil a well-known tangential condition, which is expressed by means of the Bouligand cone.

1995 - A Metric Characterization of Convex Bodies [Articolo su rivista]
Benassi, Carlo 6/8/1962; M., Boni; Gavioli, Andrea
abstract

We show that a subset C of the euclidean space which agrees with the closure of its interior is convex if and only if for everyconvex body D which meets its interior it is possibleto control, in a suitable way, the distance of a point from the intersection between C and D by means of its respectivedistances from C and D.

1994 - Approximation from the Exterior of a Multifunction with connected Values Defined on an Interval [Articolo su rivista]
Benassi, Carlo 6/8/1962; Gavioli, Andrea
abstract

We approximate an upper semicontinuous multifunction F(t) from the interval [0,1] into the compact, connected subsets of a euclidean space by means of a decreasing sequence of multifunctions which are lipschitzian with respect to the Hausdorff distance.

1994 - Approximation from the Exterior of Multifunctions with Connected Values [Articolo su rivista]
Benassi, Carlo 6/8/1962; Gavioli, Andrea
abstract

We approximate an upper semicontinuous multifunction F from a metric space T into the compact, connected subsets of a euclidean space by means of a decreasing sequence ofmultifunctions which are locally lipschitzean with respect to theHausdorff distance.

1992 - Some Bounds on the Bulk Conductivity of a two-phase Medium: a Comparison between Periodic and Random Structure [Articolo su rivista]
Gavioli, Andrea
abstract

We prove that the effective conductivity of a three-dimensional medium with a periodic chessboard structure does not exceed\Lambda\sqrt{\alpha\beta}, where \alpha and \beta are the values of the conductivity in the cells of the chessboard, and \Lambda is a positive constant; then we show how the corresponding "random" structure behaves in a quite different way, according to recent results in percolation theory.

1991 - Approximation from the exterior of a multifunction and its applications in the sweeping process [Articolo su rivista]
Gavioli, Andrea
abstract

We approximate from the exterior an upper semicontinuousmultifunction C(t) from a metric space into the closed convex subsets of a normed space by means of globally Lipschitzean multifunctions; in particular, when C(t) is continuous, this approximation allows to reduce the problem of the existence of solutions of the associated evolution equation to the case in which C(t) is Lipschitzian.

1988 - Necessary and Sufficient Conditions for the Lower Semicontinuity of Certain Integral Functionals [Articolo su rivista]
Gavioli, Andrea
abstract

We give necessary and sufficient conditions in order that a multiple integral functional of the Calculus of Variations is lower semicontinuous with respect to L^1-convergence.

1987 - Some Results about Relaxation of Integral Functionals [Articolo su rivista]
Benassi, Carlo 6/8/1962; Gavioli, Andrea
abstract

We give a representation formula for the integrand of the relaxed functional of the integral of the Calculus of Variations, in the case in which it is defined on vector functions of a real variable.

1986 - Omogeneizzazione stocastica di funzionali non coercivi [Articolo su rivista]
Facchinetti, Gisella; Gavioli, Andrea
abstract

Si fornisce un risultato di Gamma-convergenza per una successione di funzionali integrali non coercitivi associati ad un processo di omogeneizzazione stocastica. Si fa uso, allo scopo, di recenti risultati di teoria ergodica.

1982 - A Lower Semicontinuity Theorem for the Integral of the Calculus of Variations [Articolo su rivista]
Gavioli, Andrea
abstract

We prove the lower semicontinuity of a an integral functional of the kind \int_{\Omega}L(x,u(x),Du(x))dx with respect to the convercence induced by L^1_{loc} on W^{1,1}: we suppose that L(x,u,v) has an at least linear growth with respect to v and fulfils a particular property which includes some well-known cases.

1981 - An Approximation Result for Integrands of the Calculus of Variations [Articolo su rivista]
Gavioli, Andrea
abstract

We approximate from below an integrand L(t,u,v) by means of functions which enjoy Lipschitz properties with respect to u and v. Convexity on v is preserved.

1981 - On the Upper Semicontinuity of the Hamiltonian [Articolo su rivista]
Gavioli, Andrea
abstract

We give upper semicontinuity results for Fenchel's conjugate (with respect to v) of a function L(u,v).