CVGMT Papershttp://cvgmt.sns.it/papers/en-usThu, 14 Dec 2017 08:08:49 +0000On the gap between Gamma-limit and pointwise limit for a non-local approximation of the total variationhttp://cvgmt.sns.it/paper/3701/C. Antonucci, M. Gobbino, N. Picenni.<p> We consider the approximation of the total variation of a function by thefamily of non-local and non-convex functionals introduced by H. Brezis andH.-M. Nguyen in a recent paper. The approximating functionals are definedthrough double integrals in which every pair of points contributes according tosome interaction law. In this paper we answer two open questions concerning the dependence of theGamma-limit on the interaction law. In the first result, we show that theGamma-limit depends on the full shape of the interaction law, and not only onthe values in a neighborhood of the origin. In the second result, we show thatthere do exist interaction laws for which the Gamma-limit coincides with thepointwise limit on smooth functions. The key argument is that for some special classes of interaction laws thecomputation of the Gamma-limit can be reduced to studying the asymptoticbehavior of suitable multi-variable minimum problems.</p>http://cvgmt.sns.it/paper/3701/Non-uniqueness for the transport equation with Sobolev vector fieldshttp://cvgmt.sns.it/paper/3700/S. Modena, L. J. Székelyhidi.<p>We construct a large class of examples of non-uniqueness for the linear transport equation and the transport-diffusion equation with divergence-free vector fields in Sobolev spaces $W^{1,p}$.</p>http://cvgmt.sns.it/paper/3700/Multiplicative controllability for semilinear reaction-diffusion equations with finitely many changes of signhttp://cvgmt.sns.it/paper/3699/P. Cannarsa, G. Floridia, A. Y. Khapalov.<p>We study the global approximate controllability properties of a one-dimensional semilinear reaction–diffusion equation governed via the coefficient of the reaction term. It is assumed that both the initial and target states admit no more than finitely many changes of sign. Our goal is to show that any target state, with as many changes of sign in the same order as the given initial data, can be approximately reached in the L<sup>2</sup>-norm at some time T>0. Our method employs shifting the points of sign change by making use of a finite sequence of initial-value pure diffusion problems.</p>http://cvgmt.sns.it/paper/3699/Multiplicative controllability for nonlinear degenerate parabolic equations between sign-changing stateshttp://cvgmt.sns.it/paper/3698/G. Floridia, C. Nitsch, C. Trombetti.<p> In this paper we study the global approximate multiplicative controllabilityfor nonlinear degenerate parabolic Cauchy problems. In particular, we considera one-dimensional semilinear degenerate reaction-diffusion equation indivergence form governed via the coefficient of the reaction term (bilinear ormultiplicative control). The above one-dimensional equation is degenerate sincethe diffusion coefficient is positive on the interior of the spatial domain andvanishes at the boundary points. Furthermore, two different kinds of degeneratediffusion coefficient are distinguished and studied in this paper: the weaklydegenerate case, that is, if the reciprocal of the diffusion coefficient issummable, and the strongly degenerate case, that is, if that reciprocal isn'tsummable. In our main result we show that the above systems can be steered froman initial continuous state that admits a finite number of points of signchange to a target state with the same number of changes of sign in the sameorder. Our method uses a recent technique introduced for uniformly parabolicequations employing the shifting of the points of sign change by making use ofa finite sequence of initial-value pure diffusion problems. Our interest indegenerate reaction-diffusion equations is motivated by the study of someenergy balance models in climatology (see, e.g., the Budyko-Sellers model),some models in population genetics (see, e.g., the Fleming-Viot model), andsome models arising in mathematical finance (see, e.g., the Black-Scholesequation in the theory of option pricing).</p>http://cvgmt.sns.it/paper/3698/Exact controllability for quasilinear perturbations of KdVhttp://cvgmt.sns.it/paper/3697/P. Baldi, G. Floridia, E. Haus.<p>We prove that the KdV equation on the circle remains exactly controllable in arbitrary time with localized control, for sufficiently small data, also in the presence of quasilinear perturbations, namely nonlinearities containing up to three space derivatives, having a Hamiltonian structure at the highest orders. We use a procedure of reduction to constant coefficients up to order zero (adapting a result of Baldi, Berti and Montalto (2014)), the classical Ingham inequality and the Hilbert uniqueness method to prove the controllability of the linearized operator. Then we prove and apply a modified version of the Nash–Moser implicit function theorems by Hörmander (1976, 1985).</p>http://cvgmt.sns.it/paper/3697/Well-posedness of 2-D and 3-D swimming models in incompressible fluids governed by Navier-Stokes equationshttp://cvgmt.sns.it/paper/3696/P. Cannarsa, G. Floridia, A. Y. Khapalov, F. S. Priuli.<p>We introduce and investigate the wellposedness of two models describing the self-propelled motion of a “small bio-mimetic swimmer” in the 2-D and 3-D incompressible fluids modeled by the Navier–Stokes equations. It is assumed that the swimmer's body consists of finitely many subsequently connected parts, identified with the fluid they occupy, linked by the rotational and elastic forces. The swimmer employs the change of its shape, inflicted by respective explicit internal forces, as the means for self-propulsion in a surrounding medium. Similar models were previously investigated in <a href='15–19'>15–19</a> where the fluid was modeled by the liner nonstationary Stokes equations. Such models are of interest in biological and engineering applications dealing with the study and design of propulsion systems in fluids and air.</p>http://cvgmt.sns.it/paper/3696/Approximate controllability for nonlinear degenerate parabolic problems with bilinear controlhttp://cvgmt.sns.it/paper/3695/G. Floridia.<p>In this paper, we study the global approximate multiplicative controllability for nonlinear degenerate parabolic Cauchy–Neumann problems. First, we obtain embedding results for weighted Sobolev spaces, that have proved decisive in reaching well-posedness for nonlinear degenerate problems. Then, we show that the above systems can be steered in $L^2$ from any nonzero, nonnegative initial state into any neighborhood of any desirable nonnegative target-state by bilinear piecewise static controls. Moreover, we extend the above result relaxing the sign constraint on the initial data.</p>http://cvgmt.sns.it/paper/3695/Approximate multiplicative controllability for degenerate parabolic problems with Robin boundary conditionshttp://cvgmt.sns.it/paper/3694/P. Cannarsa, G. Floridia.<p>In this work we study the global approximate multiplicative controllability for a weakly degenerate parabolic Cauchy-Robin problem. The problem is weakly degenerate in the sense that the diffusion coefficient is positive in the interior of the domain and is allowed to vanish at the boundary, provided the reciprocal of the diffusion coefficient is summable. In this paper, we will show that the above system can be steered, in the space of square-summable functions, from any nonzero, nonnegative initial state into any neighborhood of any desirable nonnegative target-state by bilinear static controls. Moreover, we extend the above result relaxing the sign constraint on the initial-state.</p>http://cvgmt.sns.it/paper/3694/Approximate controllability for linear degenerate parabolic problems with bilinear controlhttp://cvgmt.sns.it/paper/3693/P. Cannarsa, G. Floridia.<p> In this work we study the global approximate multiplicative controllabilityfor the linear degenerate parabolic Cauchy-Neumann problem $$ \{{array}{l}\displaystyle{v<sub>t</sub>-(a(x) v<sub>x)</sub><sub>x</sub> =\alpha (t,x)v\,\,\qquad {in} \qquad Q<sub>T</sub>\,=\,(0,T)\times(-1,1)} <a href='2.5ex'>2.5ex</a> \displaystyle{a(x)v<sub>x</sub>(t,x)<br><sub>{x=\pm</sub> 1} =0\,\,\qquad\qquad\qquad\,\, t\in (0,T)} <a href='2.5ex'>2.5ex</a> \displaystyle{v(0,x)=v<sub>0</sub> (x)\,\qquad\qquad\qquad\qquad\quad\,\, x\in (-1,1)}, {array}. $$ with the bilinearcontrol $\alpha(t,x)\in L^\infty (Q_T).$ The problem is strongly degenerate inthe sense that $a\in C^1([-1,1]),$ positive on $(-1,1),$ is allowed to vanishat $\pm 1$ provided that a certain integrability condition is fulfilled. Wewill show that the above system can be steered in $L^2(\Omega)$ from anynonzero, nonnegative initial state into any neighborhood of any desirablenonnegative target-state by bilinear static controls. Moreover, we extend theabove result relaxing the sign constraint on $v_0$.</p>http://cvgmt.sns.it/paper/3693/DIFFERENTIABILITY AND PARTIAL HOLDER CONTINUITY OF SOLUTIONS OF NONLINEAR ELLIPTIC SYSTEMShttp://cvgmt.sns.it/paper/3692/G. Floridia, M. A. Ragusa.http://cvgmt.sns.it/paper/3692/Bending-torsion moments in thin multi-structures in the context of nonlinear elasticityhttp://cvgmt.sns.it/paper/3691/R. Ferreira, E. Zappale.<p> Here, we address a dimension-reduction problem in the context of nonlinearelasticity where the applied external surface forces induce bending-torsionmoments. The underlying body is a multi-structure in $\mathbb{R}^3$ consistingof a thin tube-shaped domain placed upon a thin plate-shaped domain. Theproblem involves two small parameters, the radius of the cross-section of thetube-shaped domain and the thickness of the plate-shaped domain. Wecharacterize the different limit models, including the limit junctioncondition, in the membrane-string regime according to the ratio between thesetwo parameters as they converge to zero.</p>http://cvgmt.sns.it/paper/3691/First-order, stationary mean-field games with congestionhttp://cvgmt.sns.it/paper/3690/Diogo A. Gomes, D. Evangelista, R. Ferreira, L. Nurbekyan, V. Voskanyan.<p> Mean-field games (MFGs) are models for large populations of competingrational agents that seek to optimize a suitable functional. In the case ofcongestion, this functional takes into account the difficulty of moving inhigh-density areas. Here, we study stationary MFGs with congestion withquadratic or power-like Hamiltonians. First, using explicit examples, weillustrate two main difficulties: the lack of classical solutions and theexistence of areas with vanishing density. Our main contribution is a newvariational formulation for MFGs with congestion. This formulation was notpreviously known, and, thanks to it, we prove the existence and uniqueness ofsolutions. Finally, we consider applications to numerical methods.</p>http://cvgmt.sns.it/paper/3690/A chromaticity-brightness model for color images denoising in a Meyer's "u + v'' frameworkhttp://cvgmt.sns.it/paper/3689/R. Ferreira, I. Fonseca, L. Mascarenhas.<p>A variational model for imaging segmentationand denoising color images is proposed. The model combines Meyer's ``u+v"decomposition with a chromaticity-brightness framework and is expressed by a minimization of energy integral functionals dependingon a small parameter $\varepsilon >0$. The asymptotic behavior as $\varepsilon\to0^+$is characterized,and convergence of infima, almost minimizers, and energies are established. In particular, an integral representation of the lower semicontinuous envelope,with respect to the $L^1$-norm, of functionals with linear growth and definedfor maps taking values on a certain compact manifold is provided. This studyescapes the realm of previous results since the underlying manifold has boundary,and the integrand and its recession function fail to satisfy hypothesescommonly assumed in the literature. The main toolsare $\Gamma$-convergence and relaxation techniques.</p>http://cvgmt.sns.it/paper/3689/Second order differentiation formula on $RCD(K,N)$ spaceshttp://cvgmt.sns.it/paper/3687/N. Gigli, L. Tamanini.<p>We prove the second order differentiation formula along geodesics in finite-dimensional $RCD(K,N)$ spaces. Our approach strongly relies on the approximation of $W_2$-geodesics by entropic interpolations and, in order to implement this approximation procedure, onthe proof of new (even in the smooth setting) estimates for such interpolations.</p>http://cvgmt.sns.it/paper/3687/Marginals with finite repulsive costhttp://cvgmt.sns.it/paper/3686/U. Bindini.<p>We consider a multimarginal transport problem with repulsive cost, where the marginals are all equal to a fixed probability $\rho \in \mathcal{P}(\mathbb{R}^d)$. We prove that, if the concentration of $\rho$ is less than $1/N$, then the problem has a solution of finite cost. The result is sharp, in the sense that there exists $\rho$ with concentration $1/N$ for which $C(\rho) = \infty$.</p>http://cvgmt.sns.it/paper/3686/Rank-one theorem and subgraphs of BV functions in Carnot groupshttp://cvgmt.sns.it/paper/3685/S. Don, A. Massaccesi, D. Vittone.<p>We prove a rank-one theorem à la G. Alberti for the derivatives of vector-valued maps with bounded variation in a class of Carnot groups that includes Heisenberg groups $\mathbb H^n$ for $n\geq 2$. The main tools are properties relating the horizontal derivatives of a real-valued function with bounded variation and its subgraph.</p>http://cvgmt.sns.it/paper/3685/Dirichlet problems for singular elliptic equations with general nonlinearitieshttp://cvgmt.sns.it/paper/3684/V. De Cicco, D. Giachetti, F. Oliva, F. Petitta.http://cvgmt.sns.it/paper/3684/Regularity results for vectorial minimizers of a class of degenerate convex integralshttp://cvgmt.sns.it/paper/3683/G. Cupini, F. Giannetti, R. Giova, A. Passarelli di Napoli.<p>We establish the higher differentiability and the higher integrability for the gradient of vec- torial minimizers of integral functionals with (p,q)-growth conditions. We assume that the non- homogeneous densities are uniformly convex and have a radial structure, with respect to the gradient variable, only at infinity. The results are obtained under a possibly discontinuous dependence on the spatial variable of the integrand.</p>http://cvgmt.sns.it/paper/3683/Willmore flow of planar networkshttp://cvgmt.sns.it/paper/3682/H. Garcke, J. Menzel, A. Pluda.<p>Geometric gradient flows for elastic energies of Willmoretype play an important role in mathematics and in many applications.The evolution of elastic curves has been studied in detail both forclosed as well as for open curves. Although elastic flows for networksalso have many interesting features, they have not been studied so farfrom the point of view of mathematical analysis. So far it was not even clear what are appropriate boundary conditions at junctions. In this paper wegive a well-posedness result for Willmore flow of networks in differentgeometric settings and hence lay a foundation for further mathematicalanalysis. A main point in the proof is to check whether different proposed boundaryconditions lead to a well posed problem. In this context one has tocheck the Lopatinskii--Shapiro condition in order to applythe Solonnikov theory for linear parabolic systems in H\"older spaceswhich is needed in a fixed point argument. We also show that the solution we get is unique in a purely geometric sense.</p>http://cvgmt.sns.it/paper/3682/An extension of the pairing theory between divergence-measure fields and BV functionshttp://cvgmt.sns.it/paper/3681/G. Crasta, V. De Cicco.<p>In this paper we introduce a nonlinear version of the notion of Anzellotti's pairingbetween divergence--measure vector fields and functions of bounded variation,motivated by possible applications toevolutionary quasilinear problems.As a consequence of our analysis, we prove a generalized Gauss--Green formula.</p>http://cvgmt.sns.it/paper/3681/