CVGMT Papershttp://cvgmt.sns.it/papers/en-usSat, 25 Nov 2017 10:18:22 -0000On the asymptotic behaviour of nonlocal perimetershttp://cvgmt.sns.it/paper/3675/J. Berendsen, V. Pagliari.
<p>We study a class of integral functionals known as nonlocal perimeters, which, intuitively, express a weighted interaction between a set and its complement. The weight is
provided by a positive kernel $K$, which might be singular.
</p>
<p>In the first part of the paper, we show that these functionals are indeed perimeters in an generalised sense and we establish existence of minimisers for the corresponding Plateau’s problem; also, when $K$ is radial and strictly decreasing, we prove that halfspaces are minimisers if we prescribe “flat” boundary conditions.
</p>
<p>A $\Gamma$-convergence result is discussed in the second part of the work. We study the limiting behaviour of the nonlocal perimeters associated with certain rescalings of a given kernel that has faster-than-$L^1$ decay at infinity and we show that the $\Gamma$-limit is the classical perimeter, up to a multiplicative constant that we compute explicitly.</p>
http://cvgmt.sns.it/paper/3675/The Pontryagin Maximum Principle in the Wasserstein Spacehttp://cvgmt.sns.it/paper/3674/B. Bonnet, F. Rossi.
<p>We prove a Pontryagin Maximum Principle for optimal control problems in the space of probability measures, where the dynamics is given by a transport equation with non-local velocity. We formulate this first-order optimality condition using the formalism of subdifferential calculus in Wasserstein spaces. We show that the geometric approach based on needle variations and on the evolution of the covector (here replaced by the evolution of a mesure on the dual space) can be translated into this formalism.</p>
http://cvgmt.sns.it/paper/3674/Sharp global estimates for local and nonlocal porous medium-type equations in bounded domainshttp://cvgmt.sns.it/paper/3673/M. Bonforte, A. Figalli, J. L. Vazquez.
<p>This paper provides a quantitative study of nonnegative solutions to nonlinear diffusion equations
of porous medium-type of the form $\partial_t u + \mathcal L u^m=0$, $m>1$, where the operator $\mathcal L$ belongs to a general class of linear operators, and the equation is posed in a bounded domain $\Omega\subset \mathbb R^N$.
As possible operators we include the three most common definitions of the fractional Laplacian in a bounded domain with zero Dirichlet conditions, and also a number of other nonlocal versions. In particular, $\mathcal L$
can be a power of a uniformly elliptic operator with $C^1$ coefficients. Since
the nonlinearity is given by $u^m$
with $m>1$, the equation is degenerate parabolic.
</p>
<p>The basic well-posedness theory for this class of equations has been recently developed in \cite{BV-PPR1,BV-PPR2-1}.
Here we address the regularity theory: decay and positivity, boundary behavior, Harnack inequalities, interior and boundary regularity, and asymptotic behavior. All this is done in a quantitative way, based on sharp a priori estimates. Although our focus is on the fractional models, our results cover also the local case when $\mathcal L$ is a uniformly elliptic operator, and provide new estimates even in this setting.
</p>
<p>A surprising aspect discovered in this paper is the possible presence of non-matching powers for the long-time boundary behavior. More precisely, when $\mathcal L=(-\Delta)^s$ is a spectral power of the {Dirichlet} Laplacian inside a smooth domain,
we can prove that:
</p>
<p>- when $2s> 1-1/m$, for large times all solutions behave as ${\rm dist}^{1/m}$ near the boundary;
</p>
<p>- when $2s\leq 1-1/m$, different solutions may exhibit different boundary behavior.
</p>
<p>This unexpected phenomenon is a completely new feature of the nonlocal nonlinear structure of this model, and it is not present in the semilinear elliptic equation $\mathcal L u^m=u$.</p>
http://cvgmt.sns.it/paper/3673/Minimal solutions to generalized $\Lambda$-semiflows and gradient flows in metric spaceshttp://cvgmt.sns.it/paper/3672/F. Fleißner.
<p>Generalized $\Lambda$-semiflows are an abstraction of semiflows with non-periodic solutions, for which there may be more than one solution corresponding to given initial data. A select class of solutions to generalized $\Lambda$-semiflows is introduced. It is proved that such minimal solutions are unique corresponding to given ranges and generate all other solutions by time reparametrization. Special qualities of minimal solutions are shown.
</p>
<p>The concept of minimal solutions is applied to gradient flows in metric spaces and generalized semiflows. Generalized semiflows have been introduced by Ball.
</p>
http://cvgmt.sns.it/paper/3672/Towards geometric integration of rough differential formshttp://cvgmt.sns.it/paper/3671/E. Stepanov, D. Trevisan.
<p>We provide a draft of a theory of geometric integration of
``rough differential forms'' which are generalizations of classical
(smooth) differential forms to similar objects with very low regularity,
for instance,
involving Hölder continuous functions that may be nowhere
differentiable. Borrowing ideas from the theory of rough paths, we show
that such a geometric integration can be constructed substituting
appropriately differentials with more general asymptotic expansions. This
can be seen as the basis of geometric integration similar to that used in
geometric measure theory, but without any underlying differentiable
structure, thus allowing Lipschitz functions and rectifiable sets to be
substituted by far less regular objects (e.g.\ Hölder functions
and their images which may be purely unrectifiable). Our
construction includes both the one-dimensional Young integral and
multidimensional integrals introduced recently by R. Züst. To simplify the
exposition, we limit ourselves to integration of forms of dimensions not
exceeding two.</p>
http://cvgmt.sns.it/paper/3671/Alexandrov theorem revisitedhttp://cvgmt.sns.it/paper/3670/M. Delgadino, F. Maggi.
<p>We show that <i>among sets of finite perimeter</i> balls are the only volume-constrained critical points of the perimeter functional. We also prove that, again among sets of finite perimeter, Wulff shapes are the only local minimizers of uniformly elliptic, smooth anisotropic surface energies.</p>
http://cvgmt.sns.it/paper/3670/A Ginzburg-Landau model with topologically induced free discontinuitieshttp://cvgmt.sns.it/paper/3669/M. Goldman, B. Merlet, V. Millot.
<p>We study a variational model which combines features of the Ginzburg-Landau model in 2D and of the Mumford-Shah functional. As in the classical Ginzburg-Landau theory,
a prescribed number of point vortices appear in the small energy regime; the model allows for discontinuities, and the energy penalizes their length. The novel phenomenon here is that the vortices have a fractional degree $1/m$ with $m\geq2$ prescribed. Those vortices must be connected by line discontinuities to form clusters of total integer degrees. The vortices and line discontinuities are therefore coupled through a topological constraint.
As in the Ginzburg-Landau model, the energy is parameterized by a small length scale $\varepsilon>0$. We perform a complete $\Gamma$-convergence analysis of the model as $\varepsilon\downarrow0$ in the small energy regime. We then study the structure of minimizers of the limit problem. In particular, we show that the line discontinuities of a minimizer solve a variant of the Steiner problem. We finally prove that for small $\varepsilon>0$, the minimizers of the original problem have the same structure away from the limiting vortices. </p>
http://cvgmt.sns.it/paper/3669/Soap film spanning an elastic linkhttp://cvgmt.sns.it/paper/3668/G. Bevilacqua, L. Lussardi, A. Marzocchi.
<p>We study the equilibrium problem of a system consisting by several Kirchhoff rods linked in an arbitrary way and tied by a soap film, using techniques of the Calculus of Variations. We prove the existence of a solution with minimum energy, which may be quite irregular, and perform experiments confirming the kind of surface predicted by the model.</p>
http://cvgmt.sns.it/paper/3668/Reverse approximation of gradient flows as Minimizing Movements: a conjecture by De Giorgihttp://cvgmt.sns.it/paper/3667/F. Fleißner, G. Savaré.
<p>We consider the Cauchy problem for a gradient ﬂow (GF) generated by a continuously diﬀerentiable function in a Hilbert space H and study the reverse approximation of its solutions by the De Giorgi Minimizing Movement approach.
</p>
<p>We prove that if H has ﬁnite dimension and the driving potential is quadratically bounded from below (in particular if it is Lipschitz) then for every solution u of (GF) (which may have an infinite number of solutions) there exist perturbations depending on the time step and converging to the potential in the Lipschitz norm such that u can be approximated by the perturbed Minimizing Movement scheme.
</p>
<p>This result solves a question raised by
E. De Giorgi, New problems on minimizing movements, in Boundary Value Problems for PDE and Applications, C. Baiocchi and J. L. Lions, eds., Masson, 1993, pp. 81–98.
</p>
<p>We also show that even if H has inﬁnite dimension the above approximation holds for the distinguished class of minimal solutions, that generate all the other solutions to the gradient flow by time reparametrization.
</p>
http://cvgmt.sns.it/paper/3667/Sensitivity of the compliance and of the Wasserstein distance with respect to a varying sourcehttp://cvgmt.sns.it/paper/3666/G. Bouchitté, I. Fragalà, I. Lucardesi.
<p>e show that the compliance functional in elasticity is differentiable with respect to horizontal variations of the load term, when the latter is given by a possibly concentrated measure; moreover,
we provide an integral representation formula for the derivative as a linear functional of the deformation vector field.
The result holds true as well for the $p$-compliance in the scalar case of conductivity.
Then we study the limit problem as $p \to + \infty$, which corresponds to differentiate the Wasserstein distance in optimal mass transportation with respect to horizontal perturbations of the two marginals. Also in this case, we obtain an existence result for the derivative, and we show that it is found by solving a minimization problem over the family of all optimal transport plans.
When the latter contains only one element, we prove that
the derivative of the $p$-compliance converges to the derivative of the Wasserstein distance in the limit as $p \to + \infty$.</p>
http://cvgmt.sns.it/paper/3666/On the curvature energy of Cartesian surfaceshttp://cvgmt.sns.it/paper/3665/D. Mucci.
<p>We analyze the lower semicontinuous envelope of the curvature functional of Cartesian surfaces in codimension one.
To this aim, following the approach by Anzellotti-Serapioni-Tamanini, we study the class of currents that naturally arise as weak limits of Gauss graphs of smooth functions.
The curvature measures are then studied in the non-parametric case. Concerning homogeneous functions, some model examples are studied in detail. Finally, a new gap phenomenon is observed.</p>
http://cvgmt.sns.it/paper/3665/Introduction to Riemannian and Sub-Riemannian geometryhttp://cvgmt.sns.it/paper/3664/A. Agrachev, D. Barilari, U. Boscain.
<p>Lecture notes "Introduction to Riemannian and Sub-Riemannian geometry".
</p>
<p><b>New updated version 17.11.2017</b> (Ch. 13 added + Revision Ch. 8, 10, 20 + New sections added in Ch. 3, 12)
</p>
<p>Table of Contents: 1 - Geometry of surfaces in R3. 2 - Vector fields and vector bundle. 3 - Sub-Riemannian structures. 4 - Characterization and local minimality of Pontryagin extremals. 5 - Integrable systems. 6 - Chronological calculus. 7 - Lie groups and left-invariant sub-Riemannian structures 8 - End-point and exponential map. 9 - 2D Almost-Riemannian structures. 10 - Nonholonomic tangent space. 11 - Regularity of the sub-Riemannian distance. 12 - Abnormal extremals and second variation. 13 - Some model spaces 14 - Curves in the Lagrange Grassmannian 15 - Jacobi curves. 16 - Riemannian curvature. 17 - Curvature of 3D contact sub-Riemannian structures. 18 - Asymptotic expansion of the 3D contact exponential map. 19 - The volume in sub-Riemannian geometry. 20 - The sub-Riemannian heat equation.</p>
http://cvgmt.sns.it/paper/3664/Some Sphere Theorems in Linear Potential Theoryhttp://cvgmt.sns.it/paper/3663/S. Borghini, G. Mascellani, L. Mazzieri.
<p> In this paper we analyze the capacitary potential due to a charged body in
order to deduce sharp analytic and geometric inequalities, whose equality cases
are saturated by domains with spherical symmetry. In particular, for a regular
bounded domain $\Omega \subset \mathbb{R}^n$, $n\geq 3$, we prove that if the
mean curvature $H$ of the boundary obeys the condition $ - \bigg[
\frac{1}{\text{Cap}(\Omega)} \bigg]^{\frac{1}{n-2}} \leq \frac{H}{n-1} \leq
\bigg[ \frac{1}{\text{Cap}(\Omega)} \bigg]^{\frac{1}{n-2}} $, then $\Omega$
is a round ball.
</p>
http://cvgmt.sns.it/paper/3663/Higher Holder regularity for the fractional $p-$Laplacian in the superquadratic casehttp://cvgmt.sns.it/paper/3662/L. Brasco, E. Lindgren, A. Schikorra.
<p>We prove higher H\"older regularity for solutions of equations involving the fractional $p-$Laplacian of order $s$,
when $p\ge 2$ and $0<s<1$. In particular, we provide an explicit H\"older exponent for solutions of the non-homogeneous equation with data in $L^q$ and $q>N/(s\,p)$,
which is almost sharp whenever $s\,p\leq (p-1)+N/q$. The result is new already for the homogeneous equation.</p>
http://cvgmt.sns.it/paper/3662/Splitting schemes & segregation in reaction-(cross-)diffusion systemshttp://cvgmt.sns.it/paper/3661/J. A. Carrillo, S. Fagioli, F. Santambrogio, M. Schmidtchen.
<p>One of the most fascinating phenomena observed in reaction-diffusion systems is the emergence of segregated solutions, \emph{i.e.} population densities with disjoint supports. We analyse such a reaction cross-diffusion system in 1D. In order to prove existence of weak solutions for a wide class of initial data without restriction about their supports or their positivity, we propose a variational splitting scheme combining ODEs with methods from optimal transport. In addition, this approach allows us to prove conservation of segregation for initially segregated data even in the presence of vacuum.
</p>
http://cvgmt.sns.it/paper/3661/A counterexample to gluing theorems for MCP metric measure spaceshttp://cvgmt.sns.it/paper/3660/L. Rizzi.
<p> Perelman's doubling theorem asserts that the metric space obtained by gluing
along their boundaries two copies of an Alexandrov space with curvature $\geq
\kappa$ is an Alexandrov space with the same dimension and satisfying the same
curvature lower bound. We show that this result cannot be extended to metric
measure spaces satisfying synthetic Ricci curvature bounds in the
$\mathrm{MCP}$ sense. The counterexample is given by the Grushin half-plane,
which satisfies the $\mathrm{MCP}(0,N)$ if and only if $N\geq 4$, while its
double satisfies the $\mathrm{MCP}(0,N)$ if and only if $N\geq 5$.
</p>
http://cvgmt.sns.it/paper/3660/Low energy configurations of topological singularities in two dimensions: A $\Gamma$-convergence analysis of dipoleshttp://cvgmt.sns.it/paper/3659/L. De Luca, M. Ponsiglione.
<p>This paper deals with the variational analysis of topological singularities in two dimensions. We consider two canonical zero-temperature models: the {core radius approach} and the Ginzburg-Landau energy.
Denoting by $\varepsilon$ the length scale parameter in such models, we focus on the $\log\frac{1}{\varepsilon}$ energy regime.
It is well known that, for configurations whose energy is bounded by $c \log \frac{1}{\varepsilon}$,
the vorticity measures can be decoupled into the sum of a finite number of Dirac masses, each one of them carrying $\pi \log \frac{1}{\varepsilon}$ energy,
plus a measure supported on small zero-average sets.
Loosely speaking, on such sets the vorticity measure is close, with respect to the flat norm, to zero-average clusters of positive and negative masses.
</p>
<p>Here we perform a compactness and $\Gamma$-convergence analysis accounting also for the presence of such clusters of dipoles (on the range scale $\varepsilon^s$, for $0<s<1$), which vanish in the flat convergence and whose energy contribution has, so far, been neglected. Our results refine and contain as a particular case the classical
$\Gamma$-convergence analysis for vortices, extending it also to low energy configurations consisting of just clusters of dipoles, and whose energy is of order $c \log \frac{1}{\varepsilon}$ with $c<\pi$.
</p>
http://cvgmt.sns.it/paper/3659/On the codimension of the abnormal set in step two Carnot groupshttp://cvgmt.sns.it/paper/3658/A. Ottazzi, D. Vittone.
<p>In this article we prove that the codimension of the abnormal set of the endpoint map for certain classes of Carnot groups of step 2 is at least three. Our result applies to all step 2 Carnot groups of dimension up to 7 and is a generalisation of a previous analogous result for step 2 free nilpotent groups. </p>
http://cvgmt.sns.it/paper/3658/Generalized crystalline evolutions as limits of flows with smooth anisotropieshttp://cvgmt.sns.it/paper/3657/A. Chambolle, M. Morini, M. Novaga, M. Ponsiglione.
<p>We prove existence and uniqueness of weak solutions to anisotropic and crystalline mean curvature flows, obtained as limit of the viscosity solutions to flows with smooth anisotropies.</p>
http://cvgmt.sns.it/paper/3657/Nonlocal problems with critical Hardy nonlinearityhttp://cvgmt.sns.it/paper/3656/W. Chen, S. J. N. Mosconi, M. Squassina.
<p>By means of variational methods we establish existence and multiplicity of solutions for a class of nonlinear nonlocal problems involving the fractional p-Laplacian and a combined Sobolev and Hardy nonlinearity at subcritical and critical growth.</p>
http://cvgmt.sns.it/paper/3656/