CVGMT Papershttp://cvgmt.sns.it/papers/en-usSat, 21 Oct 2017 16:11:15 -0000Homogenization of networks in domains with oscillating boundarieshttp://cvgmt.sns.it/paper/3641/A. Braides, V. Chiadò Piat.
<p>We consider the asymptotic behaviour of integral energies with convex integrands defined on one-dimensional networks contained in a region of the three-dimensional space with a fast-oscillating boundary as the period of the oscillation tends to zero, keeping the oscillation themselves of fixed size. The limit energy, obtained as a $\Gamma$-limit with respect to an appropriate convergence, is defined in a `stratified' Sobolev space and is written as an integral functional depending on all, two or just one derivative, depending on the connectedness properties of the sublevels of the function describing the profile of the oscillations. In the three cases, the energy function is characterized through an usual homogenization formula for $p$-connected networks, a homogenization formula for thin-film networks and a homogenization formula for thin-rod networks, respectively.
</p>
<p>This paper is dedicated to the memory of V.V.Zhikov</p>
http://cvgmt.sns.it/paper/3641/Switching mechanism in the B_{1RevTilted} phase of bent-core liquid crystalshttp://cvgmt.sns.it/paper/3640/C. J. Garcia-Cervera, T. Giorgi, S. Joo, X. Y. Lu.
http://cvgmt.sns.it/paper/3640/Error estimates for dynamic augmented Lagrangian boundary condition enforcement, with application to immersogeometric fluid–structure interactionhttp://cvgmt.sns.it/paper/3639/Y. Bazilevs, M. C. Hsu, T. J. R. Hughes, D. Kamensky, X. Y. Lu, Y. Yu.
http://cvgmt.sns.it/paper/3639/Gradient flow approach to an exponential thin film equation: global existence and latent singularityhttp://cvgmt.sns.it/paper/3638/Y. Gao, J. G. Liu, X. Y. Lu.
http://cvgmt.sns.it/paper/3638/Classification of stable solutions for boundary value problems with nonlinear boundary conditions on Riemannian manifolds with nonnegative Ricci curvaturehttp://cvgmt.sns.it/paper/3637/S. Dipierro, A. Pinamonti, E. Valdinoci.
<p>We present
a geometric formula of Poincar\'e type,
which is
inspired by a classical work of Sternberg and
Zumbrun, and
we provide a classification result
of stable solutions of linear elliptic problems
with nonlinear Robin conditions on Riemannian manifolds
with nonnegative Ricci curvature.
</p>
<p>The result obtained here is a refinement of
a result recently established by Bandle, Mastrolia,
Monticelli and Punzo.</p>
http://cvgmt.sns.it/paper/3637/The closure of planar diffeomorphisms in Sobolev spaceshttp://cvgmt.sns.it/paper/3636/G. De Philippis, A. Pratelli.
<p>We characterize the (sequentially) weak and strong closure of planar diffeomorphisms in the Sobolev topology and we show that they always coincide. We also provide some sufficient condition for a planar map to be approximable by diffeomorphisms in terms of the connectedness of its counter-images, in the spirit of Young's characterisation of monotone functions. We eventually show that the closure of diffeomorphisms in the Sobolev topology is strictly contained in the class \(INV\) introduced by Muller and Spector.</p>
http://cvgmt.sns.it/paper/3636/Confined Willmore energy and the Area functionalhttp://cvgmt.sns.it/paper/3635/M. Pozzetta.
<p>We consider minimisation problems of functionals given by the diﬀerence between the Willmore functional of a surface and its area, when the latter multiplied by a positive constant weight $\Lambda$ and when the surfaces are conﬁned in a bounded open set $\Omega\subset\mathbb{R}^3$. We give a description of the value of the inﬁma and of the convergence of minimising sequences to integer rectiﬁable varifolds in function of the parameter $\Lambda$. We also analyse some properties of these functionals and we provide some examples. Finally we prove the existence of a $C^{1,α}\cap W^{2,2}$ surface achieving the inﬁmum of the problem when the weight $\Lambda$ is suﬃciently small.</p>
http://cvgmt.sns.it/paper/3635/Erratum to: Topological equivalence of some variational problems involving distanceshttp://cvgmt.sns.it/paper/3634/G. Buttazzo, L. De Pascale, I. Fragalà.
http://cvgmt.sns.it/paper/3634/Optimal potentials for problems with changing sign datahttp://cvgmt.sns.it/paper/3633/G. Buttazzo, F. Maestre, B. Velichkov.
<p>We consider optimal control problems where the state equation is an elliptic PDE of a Schr\"odinger type, governed by the Laplace operator $-\Delta$ with the addition of a potential $V$, and the control is the potential $V$ itself, that may vary in a suitable admissible class. In a previous paper (Ref. \cite{bgrv14}) an existence result was established under a monotonicity assumption on the cost functional, which occurs if the data do not change sign. In the present paper this sign assumption is removed and the existence of an optimal potential is still valid. Several numerical simulations, made by {\tt FreeFem++}, are shown.</p>
http://cvgmt.sns.it/paper/3633/Regularity of isoperimetric regions that are close to a smooth manifoldhttp://cvgmt.sns.it/paper/3632/S. Nardulli.
<p>In this article we give some regularity results for the boundary of isoperimetric regions in a smooth complete Riemannian manifold with variable metric under suitable bounded geometry conditions, using the theory of regularity of Allard. </p>
http://cvgmt.sns.it/paper/3632/Discrete stochastic approximations of the Mumford-Shah functionalhttp://cvgmt.sns.it/paper/3631/M. Ruf.
<p> We propose a $\Gamma$-convergent discrete approximation of the Mumford-Shah
functional. The discrete functionals act on functions defined on stationary
stochastic lattices and take into account general finite differences through a
non-convex potential. In this setting the geometry of the lattice strongly
influences the anisotropy of the limit functional. Thus we can use
statistically isotropic lattices and stochastic homogenization techniques to
approximate the vectorial Mumford-Shah functional in any dimension.
</p>
http://cvgmt.sns.it/paper/3631/Crystalline evolutions in chessboard-like microstructureshttp://cvgmt.sns.it/paper/3630/A. Malusa, M. Novaga.
<p>We describe the macroscopic behavior of evolutions by crystalline curvature of
planar sets in a
chessboard-like medium,
modeled by a periodic forcing term. We show that the underlying
microstructure may produce both pinning and confinement effects on the
geometric
motion.
</p>
http://cvgmt.sns.it/paper/3630/Some minimization problems for planar networks of elastic curveshttp://cvgmt.sns.it/paper/3629/A. Dall'Acqua, A. Pluda.
<p>In this note we announce some results that will appear in $[6]$
on the minimization of the functional
$F(\Gamma)=\int_\Gamma k^2+1\,\mathrm{d}s$,
where $\Gamma$ is a network of three curves with fixed equal angles at the two junctions.
The informal description of the results is
accompanied by a partial review of the theory of elasticae
and a diffuse discussion about the onset of interesting variants of the original problem
passing from curves to networks.
The considered energy functional $F$ is given by
the elastic energy and a term that
penalize the total length of the network. We will show that penalizing the length is
tantamount to fix it.
The paper is concluded with the explicit computation of the penalized elastic energy
of the "Figure Eight",
namely the unique closed elastica with self--intersections.</p>
http://cvgmt.sns.it/paper/3629/ Generation via variational convergence of Balanced Viscosity solutions to rate-independent systemshttp://cvgmt.sns.it/paper/3628/G. A. Bonaschi, R. Rossi.
<p>In this paper we investigate the origin of the Balanced Viscosity solution concept for rate-independent evolution in the setting of a finite-dimensional space. Namely, given a family of dissipation potentials $(\Psi_n)_n$ with superlinear growth at infinity and a smooth energy functional $\mathcal{E}$, we enucleate sufficient conditions on them ensuring that the associated gradient
systems $(\Psi_n,\mathcal{E})$ Evolutionary Gamma-converge to a limiting rate-independent system, understood in the sense of Balanced Viscosity solutions. In particular, our analysis encompasses both the vanishing-viscosity approximation of rate-independent systems
and their stochastic derivation. </p>
http://cvgmt.sns.it/paper/3628/On a Minkowski geometric flow in the planehttp://cvgmt.sns.it/paper/3627/S. Dipierro, M. Novaga, E. Valdinoci.
<p>We consider a planar geometric
flow in which the normal velocity is a
nonlocal variant of the curvature. The
flow is not scaling invariant and in fact
has different behaviors at different spatial scales, thus producing phenomena
that are different with respect to both the classical mean curvature
flow and
the fractional mean curvature
flow.
In particular, we give examples of neckpinch singularity formation, we
show that sets with "sufficiently small interior" remain convex under the
flow, but, on the other hand, in general the
flow does not preserve convexity.
We also take into account traveling waves for this geometric
flow, showing
that a new family of $C^2$ and convex traveling sets arises in this setting.</p>
http://cvgmt.sns.it/paper/3627/Spatial rational expectations equilibria in the Ramsey model of optimal growthhttp://cvgmt.sns.it/paper/3626/F. Santambrogio, A. Xepapadeas, A. Yannacopoulos.
<p>t is the aim of this work provide a rigorous treatment concerning the formation of spatial rational expectations equlibria in a general class of spatial economic models under the effect of externalities, using techniques from the calculus of variations. Using detailed estimates for a parametric optimisation problem, the existence of spatial rational expectations equilibria is proved and they are characterised in terms of a nonlocal Euler-Lagrange equation.
</p>
http://cvgmt.sns.it/paper/3626/On the singular local limit for conservation laws with nonlocal fluxeshttp://cvgmt.sns.it/paper/3625/M. Colombo, G. Crippa, L. V. Spinolo.
<p>We give an answer to a question posed in $[$P. Amorim, R. Colombo, A. Teixeira, ESAIM Math. Model. Numerics. Anal. 2015$]$, which can be loosely speaking formulated as follows. Consider a family of continuity equations where the velocity field is given by the convolution of the solution with a regular kernel. In the singular limit where the convolution kernel is replaced by a Dirac delta, one formally recovers a conservation law: can we rigorously justify this formal limit?
We exhibit counter-examples showing that, despite numerical evidence suggesting a positive answer, one in general does not have convergence of the solutions. We also show that the answer is positive if we consider viscous perturbations of the nonlocal equations. In this case, in the singular local limit the solutions converge to the solution of the viscous conservation law. </p>
http://cvgmt.sns.it/paper/3625/On Monge's problem for Bregman-like cost functions http://cvgmt.sns.it/paper/3624/G. Carlier, C. Jimenez.
<p>Clin d'oeil à Filippo Santambrogio (il sait pourquoi)</p>
http://cvgmt.sns.it/paper/3624/Completion of the space of measures in the Kantorovich normhttp://cvgmt.sns.it/paper/3623/G. Bouchitté, T. Champion, C. Jimenez.
http://cvgmt.sns.it/paper/3623/Asymptotique d'un problème de positionnement optimal.http://cvgmt.sns.it/paper/3622/G. Bouchitté, C. Jimenez, R. Mahadevan.
http://cvgmt.sns.it/paper/3622/