CVGMT Papershttp://cvgmt.sns.it/papers/en-usMon, 25 Sep 2017 10:01:59 -0000Minimizers for a fractional Allen-Cahn equation in a periodic mediumhttp://cvgmt.sns.it/paper/3590/D. Pagliardini.
<p>We aim to study the solutions of a fractional mesoscopic model of phase transitions in a periodic medium. After investigating the geometric properties of the interface of the associated minimal solutions, we construct minimal interfaces lying to a strip of prescribed direction and universal width.</p>
http://cvgmt.sns.it/paper/3590/Some characterizations of magnetic Sobolev spaceshttp://cvgmt.sns.it/paper/3589/H. M. Nguyen, A. Pinamonti, M. Squassina, E. Vecchi.
<p>The aim of this note is to survey recent results where
the authors extended to the magnetic setting several characterizations of Sobolev and BV functions.</p>
http://cvgmt.sns.it/paper/3589/Rigidity results for elliptic boundary value problemshttp://cvgmt.sns.it/paper/3588/S. Dipierro, A. Pinamonti, E. Valdinoci.
<p>We provide a general approach to the classification results
of stable solutions of (possibly nonlinear) elliptic problems
with Robin conditions.
</p>
<p>The method is based on a geometric formula of Poincar\'e type,
which is
inspired by a classical work of Sternberg and
Zumbrun and which
gives an accurate description of the curvatures
of the level sets of the stable solutions. {F}rom this, we show that the stable solutions
of a quasilinear problem with Neumann data are necessarily constant.
</p>
<p>As a byproduct of this, we obtain an alternative proof of
a celebrated result of Casten and Holland, and Matano.
</p>
<p>In addition, we will obtain as a consequence a new proof of
a result recently established by Bandle, Mastrolia,
Monticelli and Punzo.</p>
http://cvgmt.sns.it/paper/3588/Multiple solutions for a self-consistent Dirac equation in two dimensionshttp://cvgmt.sns.it/paper/3587/W. Borrelli.
<p>This paper is devoted to the variational study of an effective model for the electron transport in a graphene sample. We prove the existence of infinitely many stationary solutions for a nonlinear Dirac equation which appears in the WKB limit for the Schroedinger equation describing the semi-classical electron dynamics. The interaction term is given by a mean field, self-consistent potential which is the trace of the 3D Coulomb potential. Despite the nonlinearity being 4-homogeneous, compactness issues related to the limiting Sobolev embedding $H^{\frac{1}{2}}(\Omega,\mathbb{C}^{2})\hookrightarrow L^{4}(\Omega,\mathbb{C}^{2})$ are avoided thanks to the regularization property of the operator $(-\Delta)^{-\frac{1}{2}}$. This also allows us to prove smoothness of the solutions. Our proof follows by direct arguments.</p>
http://cvgmt.sns.it/paper/3587/Isoperimetric problems for a nonlocal perimeter of Minkowski typehttp://cvgmt.sns.it/paper/3586/A. Cesaroni, M. Novaga.
<p>We prove a quantitative version of the isoperimetric inequality
for a non local perimeter of Minkowski type. We also apply this result to
study isoperimetric problems with repulsive interaction terms, under convexity
constraints. We show existence of minimizers, and we describe the shape of
minimizers in certain parameter regimes.</p>
http://cvgmt.sns.it/paper/3586/Stationary solutions for the 2D critical Dirac equation with Kerr nonlinearityhttp://cvgmt.sns.it/paper/3585/W. Borrelli.
<p>In this paper we prove the existence of an exponentially localized stationary
solution for a two-dimensional cubic Dirac equation. It appears as an
effective equation in the description of nonlinear waves for some Condensed
Matter (Bose-Einstein condensates) and Nonlinear Optics (optical bers)
systems. The nonlinearity is of Kerr-type, that is of the form $\vert\psi\vert^{2}\psi$ and
thus not Lorenz-invariant. We solve compactness issues related to the critical
Sobolev embedding $H^{\frac{1}{2}}(\mathbb{R}^{2};\mathbb{C}^{2})\hookrightarrow L^{4}(\mathbb{R}^{2};\mathbb{C}^{2})$ thanks to a particular
radial ansatz. Our proof is then based on elementary dynamical systems
arguments.</p>
http://cvgmt.sns.it/paper/3585/Equilibrium shapes of charged liquid droplets and related problems: (mostly) a reviewhttp://cvgmt.sns.it/paper/3578/M. Goldman, B. Ruffini.
<p>We review some recent results on the equilibrium shapes of charged liquid
drops. We show that the natural variational model is ill-posed and how this can be
overcome by either restricting the class of competitors or by adding penalizations in the
functional. The original contribution of this note is twofold. First, we prove existence of
an optimal distribution of charge for a conducting drop subject to an external electric
eld. Second, we prove that there exists no optimal conducting drop in this setting.</p>
http://cvgmt.sns.it/paper/3578/Dimension reduction in the context of structured deformationshttp://cvgmt.sns.it/paper/3577/G. Carita, J. Matias, M. Morandotti, D. R. Owen.
<p>In this paper we apply both the procedure of dimension reduction and the incorporation of structured deformations to a three-dimensional continuum in the form of a thinning domain.
We apply the two processes one after the other, exchanging the order, and so obtain
for each order both a relaxed bulk and a relaxed interfacial energy.
Our implementation requires some substantial modifications of the two relaxation procedures.
For the specific choice of an initial energy including only the surface term, we compute the energy densities explicitly and show that they are the same, independent of the order of the relaxation processes.
Moreover, we compare our explicit results with those obtained when the limiting process of dimension reduction and of passage to the structured deformation is carried out at the same time.
We finally show that, in a portion of the common domain of the relaxed energy densities, the simultaneous procedure gives an energy strictly lower than that obtained in the two-step relaxations.
</p>
http://cvgmt.sns.it/paper/3577/Hamilton Jacobi Isaacs equations for Differential Games with asymmetric information on probabilistic initial conditionhttp://cvgmt.sns.it/paper/3576/C. Jimenez, M. Quincampoix.
<p>We investigate Hamilton Jacobi Isaacs equations associated to a two-players zero-sum differential game with incomplete information. The first player has complete information on the initial state of the game while the second player has only information of a - possibly uncountable - probabilistic nature: he knows a probability measure on the initial state. Such differential games with finite type incomplete information can be viewed as a generalization of the famous Aumann-Maschler theory for repeated games. The main goal and novelty of the present work consists in obtaining and investigating a Hamilton Jacobi Isaacs Equation satisfied by the upper and the lower values of the game. Since we obtain a uniqueness result for such Hamilton Jacobi equation, as a byproduct, this gives an alternative proof of the existence of a value of the differential game (which has been already obtained in the literature by different technics). Since the Hamilton Jacobi equation is naturally stated in the space of probability measures, we use the Wasserstein distance and some tools of optimal transport theory.
</p>
<p></p>
http://cvgmt.sns.it/paper/3576/$L^\infty$ estimates for the JKO scheme in parabolic-elliptic Keller-Segel systemshttp://cvgmt.sns.it/paper/3575/J. A. Carrillo, F. Santambrogio.
<p>We prove $L^\infty$ estimates on the densities that are obtained via the JKO scheme for a general form of a parabolic-elliptic Keller-Segel type system, with arbitrary diffusion, arbitrary mass, and in arbitrary dimension. Of course, such an estimate blows up in finite time, a time proportional to the inverse of the initial $L^\infty$ norm. This estimate can be used to prove short-time well-posedness for a number of equations of this form regardless of the mass of the initial data. The time of existence of the constructed solutions coincides with the maximal time of existence of Lagrangian solutions without the diffusive term by characteristic methods.</p>
http://cvgmt.sns.it/paper/3575/Fractional Sobolev Spaces and Functions of Bounded Variation of One Variablehttp://cvgmt.sns.it/paper/3574/M. Bergounioux, A. Leaci, G. Nardi, F. Tomarelli.
<p>We investigate the 1D Riemann-Liouville fractional derivative focusing on the connections with fractional Sobolev spaces, the space $BV$ of functions of bounded variation, whose derivatives are not functions but measures and the space $SBV$, say the space of bounded variation functions whose derivative has no Cantor part.
We prove that $SBV$ is included in $W^{s,1} $ for every $s \in (0,1)$ while the result remains open for $BV$. We study examples and address open questions.</p>
<p>There is a minor change in the title with respect to the arxiv preprint.</p>
http://cvgmt.sns.it/paper/3574/A kinetic selection principle for curl-free vector fields of unit normhttp://cvgmt.sns.it/paper/3573/P. Bochard, P. Pegon.
<p>This article is devoted to the generalization of results obtained in 2002 by Jabin, Otto and Perthame. In their article they proved that planar vector fields taking value into the unit sphere of the euclidean norm and satisfying a given kinetic equation are locally Lipschitz. Here, we study the same question replacing the unit sphere of the euclidean norm by the unit sphere of any norm. Under natural assumptions on the norm, namely smoothness and a qualitative convexity property, that is to be of power type $p$, we prove that planar vector fields taking value into the unit sphere of such a norm and satisfying a certain kinetic equation are locally $\frac{1}{p-1}$-Hölder continuous. Furthermore we completely describe the behaviour of such a vector field around singular points as a vortex associated to the norm. As our kinetic equation implies for the vector field to be curl-free, this can be seen as a selection principle for curl-free vector fields valued in spheres of general norms which rules out line-like singularities.
</p>
http://cvgmt.sns.it/paper/3573/Fokker-Planck-Kolmogorov operators in dimension two: heat kernel and curvaturehttp://cvgmt.sns.it/paper/3572/D. Barilari, F. Boarotto.
<p>We consider the heat equation associated with a class of hypoelliptic operators of Fokker-Planck-Kolmogorov type in dimension two. We explicitly compute the first meaningful coefficient of the small time asymptotic expansion of the heat kernel on the diagonal, and we interpret it in terms of curvature-like invariants of the optimal control problem associated with the diffusion. This gives a first example of geometric interpretation of the small-time heat kernel asymptotics associated with non-homogeneous H ̈ormander operators which are not associated with a sub-Riemannian structure, i.e., whose second-order part does not satisfy the H ̈ormander condition.</p>
http://cvgmt.sns.it/paper/3572/Metric methods for heteroclinic connections in infinite dimensional spaceshttp://cvgmt.sns.it/paper/3571/A. Monteil, F. Santambrogio.
<p>We consider the minimal action problem $\min\int_\mathbb{R} (\frac{1}{2}<br>\dot{\gamma}<br>^2+W(\gamma))d t $ among curves lying in a non-locally-compact metric space and connecting two given zeros of $W\geq 0$. For this problem, the optimal curves are usually called heteroclinic connections.
We reduce it, following a standard method, to a geodesic problem of the form $\min\int_0^1 K(\gamma)\vert\dot{\gamma}\vert d t$ with $K=\sqrt{2W}$. We then prove existence of curves minimizing this new action under some suitable compactness assumptions on $K$, which are minimal. The method allows to solve some PDE problems in unbounded domains, in particular in two variables $x,y$, when $y=t$ and when the metric space is an $L^2$ space in the first variable $x$, and the potential $W$ includes a Dirichlet energy in the same variable. We then apply this technique to the problem of connecting, in a functional space, two different heteroclinic connections between two points of the Euclidean space, as it was previously studied by Alama-Bronsard-Gui and by Schatzman more than fifteen years ago. With a very different technique, we are able to recover the same results, and to weaken some assumptions.
</p>
http://cvgmt.sns.it/paper/3571/Thin films with many small crackshttp://cvgmt.sns.it/paper/3570/K. Bhattacharya, A. Braides.
<p>We show with an example that the limiting theory as thickness goes to zero of a thin film with many small cracks can be three-dimensional rather than two-dimensional.
</p>
http://cvgmt.sns.it/paper/3570/An epiperimetric inequality for the lower dimensional obstacle problemhttp://cvgmt.sns.it/paper/3569/F. Geraci.
<p>In this paper we give a proof of an epiperimetric inequality in the setting
of the lower dimensional obstacle problem. The inequality was introduced by Weiss (Invent. Math., 138 (1999), no. 1, 23-50) for the classical obstacle
problem and has striking consequences concerning the regularity of the
free-boundary. Our proof follows the approach of Focardi and Spadaro (Adv.
Differential Equations 21 (2015), no 1-2, 153-200.) which uses an homogeneity
approach and a $\Gamma$-convergence analysis.</p>
http://cvgmt.sns.it/paper/3569/Delayed loss of stability in singularly perturbed finite-dimensional gradient flowshttp://cvgmt.sns.it/paper/3568/G. Scilla, F. Solombrino.
<p>In this paper we study the singular vanishing-viscosity limit of a gradient flow in a finite dimensional Hilbert space, focusing on the so-called delayed loss of stability of stationary solutions. We find a class of time-dependent energy functionals and initial conditions for which we can explicitly calculate the first discontinuity time $t^*$ of the limit. For our class of functionals, $t^*$ coincides with the blow-up time of the solutions of the
linearized system around the equilibrium, and is in particular strictly greater than the time $t_c$ where strict local minimality with respect to the driving energy gets lost. Moreover, we show that, in a right neighborhood of $t^*$, rescaled solutions of the singularly perturbed problem converge to heteroclinic solutions of the gradient flow. Our results complement the previous ones by Zanini, where the situation we consider was excluded by assuming the so-called transversality conditions, and the limit evolution consisted of strict local minimizers of the energy up to a negligible set of times.</p>
http://cvgmt.sns.it/paper/3568/Ill-posedness of Leray solutions for the ipodissipative Navier--Stokes equationshttp://cvgmt.sns.it/paper/3567/M. Colombo, C. De Lellis, L. De Rosa .
<p>We prove the ill-posedness of Leray solutions to the Cauchy problem for the ipodissipative Navier--Stokes equations,
when the dissipative term is a fractional Laplacian $(-\Delta)^\alpha$ with exponent $\alpha < \frac{1}{5}$. The proof follows the
``convex integration methods'' introduced by the second author and L\'aszl\'o Sz\'ekelyhidi Jr. for the incomprresible Euler equations.
The methods yield indeed some conclusions even for exponents in the range $[\frac{1}{5}, \frac{1}{2}[$. </p>
http://cvgmt.sns.it/paper/3567/Fractional Sobolev regularity for the Brouwer degreehttp://cvgmt.sns.it/paper/3566/C. De Lellis, D. Inauen.
<p>We prove that if $\Omega\subset \mathbb R^n$ is a bounded open set and $n\alpha> {\rm dim}_b (\partial \Omega) = d$, then the Brouwer degree deg$(v,\Omega,\cdot)$ of any H\"older function $v\in C^{0,\alpha}\left (\Omega, {\mathbb R}^{n}\right)$ belongs to the Sobolev space $W^{\beta, p} (\mathbb R^n)$ for every $0\leq \beta < \frac{n}{p} - \frac{d}{\alpha}$. This extends a summability result of Olbermann and in fact we get, as a byproduct, a more elementary proof of it. Moreover we show the optimality of the range of exponents in the following sense: for every $\beta\geq 0$ and $p\geq 1$ with $\beta > \frac{n}{p} - \frac{n-1}{\alpha}$ there is a vector field $v\in C^{0, \alpha} (B_1, \mathbb R^n)$ with
$\mbox{deg}\, (v, \Omega, \cdot)\notin W^{\beta, p}$, where $B_1 \subset \mathbb R^n$ is the unit ball. </p>
http://cvgmt.sns.it/paper/3566/Two examples of minimal Cheeger sets in the planehttp://cvgmt.sns.it/paper/3565/G. P. Leonardi, G. Saracco.
<p> We construct two minimal Cheeger sets in the Euclidean plane, i.e. unique
minimizers of the ratio "perimeter over area" among their own measurable
subsets. The first one gives a counterexample to the so-called weak regularity
property of Cheeger sets, as its perimeter does not coincide with the
$1$-dimensional Hausdorff measure of its topological boundary. The second one
is a kind of porous set, whose boundary is not locally a graph at many of its
points, yet it is a weakly regular open set admitting a unique (up to vertical
translations) non--parametric solution to the prescribed mean curvature
equation, in the extremal case corresponding to the capillarity for perfectly
wetting fluids in zero gravity.
</p>
http://cvgmt.sns.it/paper/3565/