CVGMT Papershttp://cvgmt.sns.it/papers/en-usSun, 20 Aug 2017 23:00:22 -0000Sobolev regularity for first order Mean Field Gameshttp://cvgmt.sns.it/paper/3552/P. J. Graber, A. R. Mészáros.
<p>In this paper we obtain Sobolev estimates for weak solutions of first oder variational Mean Field Game systems with coupling terms that are local function of the density variable. Under some coercivity condition on the coupling, we obtain first order Sobolev estimates for the density variable, while under similar coercivity condition on the Hamiltonian we obtain second order Sobolev estimates for the value function. These results are valid both for stationary and time-dependent problems. In the latter case the estimates are fully global in time, thus we resolve a question which was left open in a recent paper of Prosinski and Santambrogio. Our methods apply to a large class of Hamiltonians and coupling functions.</p>
http://cvgmt.sns.it/paper/3552/Global well-posedness of the spatially homogeneous Kolmogorov-Vicsek model as a gradient flowhttp://cvgmt.sns.it/paper/3551/A. Figalli, M. J. Kang, J. Morales.
<p>We consider the so-called spatially homogenous Kolmogorov-Vicsek model,
a non-linear Fokker-Planck equation of self-driven stochastic particles with orientation
interaction under the space-homogeneity. We prove the global existence and uniqueness of
weak solutions to the equation. We also show that weak solutions exponentially converge
to a steady state, which has the form of the Fisher-von Mises distribution.</p>
http://cvgmt.sns.it/paper/3551/The ground state of long-range Schroedinger equations and static $q\overline{q}$ potentialhttp://cvgmt.sns.it/paper/3550/M. Beccaria, G. Metafune, D. Pallara.
http://cvgmt.sns.it/paper/3550/On some generalisations of Meyers-Serrin Theoremhttp://cvgmt.sns.it/paper/3549/D. Guidetti, B. Güneysu, D. Pallara.
http://cvgmt.sns.it/paper/3549/Regularity estimates for scalar conservation laws in one space dimensionhttp://cvgmt.sns.it/paper/3548/E. Marconi.
http://cvgmt.sns.it/paper/3548/Numerical solution of a nonlinear eigenvalue problem arising in optimal insulationhttp://cvgmt.sns.it/paper/3547/S. Bartels, G. Buttazzo.
<p>The optimal insulation of a heat conducting body by a thin film of variable thickness can be formulated as a nondifferentiable, nonlocal eigenvalue problem. The discretization and iterative solution for the reliable computation of corresponding eigenfunctions that determine the optimal layer thickness are addressed. Corresponding numerical experiments confirm the theoretical observation that a symmetry breaking occurs for the case of small available insulation masses and provide insight in the geometry of optimal films. An experimental shape optimization indicates that convex bodies with one axis of symmetry have favorable insulation properties.</p>
http://cvgmt.sns.it/paper/3547/Gausson dynamics for logarithmic Schrodinger equationshttp://cvgmt.sns.it/paper/3546/A. H. Ardila, M. Squassina.
<p>In this paper we study the validity of a Gausson (soliton) dynamics of the logarithmic Schrodinger equation in presence of a smooth external potential.</p>
http://cvgmt.sns.it/paper/3546/A density result in $GSBD^p$ with applications to the approximation of brittle fracture energieshttp://cvgmt.sns.it/paper/3545/A. Chambolle, V. Crismale.
<p> We prove that any function in $GSBD^p(\Omega)$, with $\Omega$ a
$n$-dimensional open bounded set with finite perimeter, is approximated by
functions $u_k\in SBV(\Omega;\mathbb{R}^n)\cap L^\infty(\Omega;\mathbb{R}^n)$ whose jump is a
finite union of $C^1$ hypersurfaces. The approximation takes place in the sense
of Griffith-type energies $\int_\Omega W(e(u)) \,\mathrm{d}x +\mathcal{H}^{n-1}(J_u)$, $e(u)$ and $J_u$
being the approximate symmetric gradient and the jump set of $u$, and $W$ a
nonnegative function with $p$-growth, $p>1$. The difference between $u_k$ and
$u$ is small in $L^p$ outside a sequence of sets $E_k\subset \Omega$ whose
measure tends to 0 and if $<br>u<br>^r \in L^1(\Omega)$ with $r\in (0,p]$, then
$<br>u_k-u<br>^r \to 0$ in $L^1(\Omega)$. Moreover, an approximation property for the
(truncation of the) amplitude of the jump holds. We apply the density result to
deduce $\Gamma$-convergence approximation \emph{\`a la} Ambrosio-Tortorelli for
Griffith-type energies with either Dirichlet boundary condition or a mild
fidelity term, such that minimisers are \emph{a priori} not even in
$L^1(\Omega;\mathbb{R}^n)$.</p>
http://cvgmt.sns.it/paper/3545/Analytical validation of the Young-Dupré law for epitaxially-strained thin filmshttp://cvgmt.sns.it/paper/3544/E. Davoli, P. Piovano.
<p>A variational model for epitaxially-strained thin films on substrates is derived both by $\Gamma$-convergence from a transition-layer setting, and by relaxation of a sharp-interface description. The model is characterized by a configurational energy that accounts for possibly different elastic properties for the film and the substrate, as well as for the surface tensions of all three involved interfaces: film-gas, substrate-gas, and film-substrate. Minimal configurations of this energy are then shown to exist and their regularity and geometrical properties are studied. The Young-Dupré law is shown to be satisfied by the angle that energetically-optimal profiles form at contact points with the substrate. This appears to be the first analytical validation of such relation, which was originally formulated in Fluid Mechanics, in the context of Continuum Mechanics for a thin-film model.</p>
http://cvgmt.sns.it/paper/3544/Rectifiability of the singular set of multiple valued energy minimizing harmonic mapshttp://cvgmt.sns.it/paper/3543/J. Hirsch, S. Stuvard, D. Valtorta.
<p> In this paper we study the singular set of Dirichlet-minimizing $Q$-valued
maps from $\mathbb{R}^m$ into a smooth compact manifold $\mathcal{N}$ without
boundary. Similarly to what happens in the case of single valued minimizing
harmonic maps, we show that this set is always $(m-3)$-rectifiable with uniform
Minkowski bounds. Moreover, as opposed to the single valued case, we prove that
the target $\mathcal{N}$ being non-positively curved but not simply connected
does not imply continuity of the map.
</p>
http://cvgmt.sns.it/paper/3543/Non-collapsed spaces with Ricci curvature bounded from belowhttp://cvgmt.sns.it/paper/3542/G. De Philippis, N. Gigli.
<p>We propose a definition of non-collapsed space with Ricci curvature bounded from below and we prove the versions of Colding's volume convergence theorem and of Cheeger-Colding dimension gap estimate for ${\sf RCD}$ spaces.
</p>
<p>In particular this establishes the stability of non-collapsed spaces under non-collapsed Gromov-Hausdorff convergence. </p>
http://cvgmt.sns.it/paper/3542/A logarithmic epiperimetric inequality for the obstacle problemhttp://cvgmt.sns.it/paper/3541/M. Colombo, L. Spolaor, B. Velichkov.
<p>For the general obstacle problem, we prove by direct methods an epiperimetric inequality at regular and singular points, thus answering a question of Weiss (Invent. Math., 138 (1999), 23–50). In particular, at singular points we introduce a new tool, which we call logarithmic epiperimetric inequality, which yields an explicit logarithmic modulus of continuity on the $C^1$ regularity of the singular set, thus improving previous results of Caffarelli and Monneau.</p>
http://cvgmt.sns.it/paper/3541/Optimal constants for a non-local approximation of Sobolev norms and
total variationhttp://cvgmt.sns.it/paper/3539/C. Antonucci, M. Gobbino, M. Migliorini, N. Picenni.
<p> We consider the family of non-local and non-convex functionals proposed and
investigated by J. Bourgain, H. Brezis and H.-M. Nguyen in a series of papers
of the last decade. It was known that this family of functionals
Gamma-converges to a suitable multiple of the Sobolev norm or the total
variation, depending on the summability exponent, but the exact constants and
the structure of recovery families were still unknown, even in dimension one.
We prove a Gamma-convergence result with explicit values of the constants in
any space dimension. We also show the existence of recovery families consisting
of smooth functions with compact support.
The key point is reducing the problem first to dimension one, and then to a
finite combinatorial rearrangement inequality.
</p>
http://cvgmt.sns.it/paper/3539/Symmetry-breaking in a generalized Wirtinger inequalityhttp://cvgmt.sns.it/paper/3540/M. Ghisi, M. Gobbino, G. Rovellini.
<p> The search of the optimal constant for a generalized Wirtinger inequality in
an interval consists in minimizing the $p$-norm of the derivative among all
functions whose $q$-norm is equal to~1 and whose $(r-1)$-power has zero
average. Symmetry properties of minimizers have attracted great attention in
mathematical literature in the last decades, leading to a precise
characterization of symmetry and asymmetry regions.
In this paper we provide a proof of the symmetry result without computer
assisted steps, and a proof of the asymmetry result which works as well for
local minimizers. As a consequence, we have now a full elementary description
of symmetry and asymmetry cases, both for global and for local minima.
Proofs rely on appropriate nonlinear variable changes.
</p>
http://cvgmt.sns.it/paper/3540/Rigidity and trace properties of divergence-measure vector fieldshttp://cvgmt.sns.it/paper/3538/G. P. Leonardi, G. Saracco.
<p>We show some rigidity properties of divergence--free vector fields defined on half--spaces. As an application, we prove the existence of the classical trace for a bounded, divergence--measure vector field $\xi$ defined on the Euclidean plane, at almost every point of a locally oriented rectifiable set $S$, under the assumption that its weak normal trace $[\xi\cdot \nu_S]$ attains a local maximum for the norm of $\xi$ at the point. </p>
http://cvgmt.sns.it/paper/3538/Anzellotti's pairing theory and the Gauss-Green theoremhttp://cvgmt.sns.it/paper/3537/G. Crasta, V. De Cicco.
<p>In this paper we obtain a very general Gauss-Green formula
for weakly differentiable functions
and sets of finite perimeter.
This result is obtained by revisiting the Anzellotti's pairing theory and by characterizing the measure
pairing (A, Du) when
A is a bounded divergence measure vector field and
u is a bounded function of bounded variation.
</p>
http://cvgmt.sns.it/paper/3537/Ancient solutions of superlinear heat equations on Riemannian manifoldshttp://cvgmt.sns.it/paper/3536/D. Castorina, C. Mantegazza.
<p>We study some qualitative properties of ancient solutions of superlinear heat equations on a Riemannian manifold, with particular interest in positivity and constancy in space.
</p>
http://cvgmt.sns.it/paper/3536/Uniqueness of solutions in Mean Field Games with several populations and Neumann conditionshttp://cvgmt.sns.it/paper/3535/M. Bardi, M. Cirant.
<p>We study the uniqueness of solutions to systems of PDEs arising in Mean Field Games with several populations of agents and Neumann boundary conditions. The main assumption requires the smallness of some data, e.g., the length of the time horizon. This complements the existence results for MFG models of segregation phenomena introduced by the authors and Achdou. An application to robust Mean Field Games is also given.</p>
http://cvgmt.sns.it/paper/3535/On non-uniqueness and uniqueness of solutions in finite-horizon Mean Field Gameshttp://cvgmt.sns.it/paper/3534/M. Bardi, M. Fischer.
<p>This paper presents a class of evolutive Mean Field Games with multiple solutions for all time horizons $T$ and convex but non-smooth Hamiltonian $H$, as well as for smooth $H$ and $T$ large enough. The phenomenon is analysed in both the PDE and the probabilistic setting. The examples are compared with the current theory about uniqueness of solutions. In particular, a new result on uniqueness for the MFG PDEs with small data, e.g., small $T$, is proved. Some results are also extended to MFGs with two populations.</p>
http://cvgmt.sns.it/paper/3534/Sweeping processes with prescribed behaviour on jumpshttp://cvgmt.sns.it/paper/3533/V. Recupero, F. Santambrogio.
<p>We present a generalized formulation of sweeping process where the behaviour of the solution is prescribed at the jump
points of the driving moving set. An existence and uniqueness theorem for such formulation is proved. As a consequence we derive a formulation and an existence<i>uniqueness theorem for sweeping processes driven by an arbitrary BV moving set, whose evolution is not necessarily right continuous. Applications to the play operator of elastoplasticity are also shown.</i></p>
http://cvgmt.sns.it/paper/3533/