cvgmt Papershttp://cvgmt.sns.it/papers/en-usSat, 18 Aug 2018 06:55:05 +0000Regularity of interfaces in phase transitions via obstacle problemshttp://cvgmt.sns.it/paper/4003/A. Figalli.<p>The aim of this note is to review some recent developments on the regularity theory for the stationary and parabolic obstacle problems. </p><p>After a general overview, we present some recent results on the structure of singular free boundary points. Then, we show some selected applications to the generic smoothness of the free boundary in the stationary obstacle problem (Schaeffer's conjecture), and to the smoothness of the free boundary in the one-phase Stefan problem for almost every time.</p>http://cvgmt.sns.it/paper/4003/A variational method for second order shape derivativeshttp://cvgmt.sns.it/paper/4002/G. Bouchitté, I. Fragalà, I. Lucardesi.<p>We consider shape functionals obtained as minima on Sobolev spaces of classical integrals having smooth and convex densities, under mixed Dirichlet--Neumann boundary conditions. We propose a new approach for the computation of the second order shape derivative of such functionals, yielding a general existence and representation theorem. In particular, we consider the $p$-torsional rigidity functional for $p \geq 2$.</p>http://cvgmt.sns.it/paper/4002/Concentration phenomena in the optimal design of thin rodshttp://cvgmt.sns.it/paper/4001/I. Lucardesi.<p>In this paper we analyze the concentration phenomena which occur in thin rods, solving thefollowing optimization problem: a given fraction of elastic material must be distributed into acylindrical design region with infinitesimal cross section in an optimal way, so that it maximizesthe resistance to a given external load. For small volume fractions, the optimal configuration ofmaterial is described by a measure which concentrates on 2-rectifiable sets. For some choices ofthe external charging, the concentration phenomena turn out to be related to some new variantsof the Cheeger problem of the cross section of the rod. The same study has already been carriedout in the particular case of pure torsion regime in <a href='G. Bouchitté, I. Fragalà, I. Lucardesi, P. Seppecher: Optimal thin torsion rods and Cheeger sets, SIAM J. Math. Anal. 44 (2012)'>G. Bouchitté, I. Fragalà, I. Lucardesi, P. Seppecher: Optimal thin torsion rods and Cheeger sets, SIAM J. Math. Anal. 44 (2012)</a>. Here we extend those results by enlarging the class of admissible loads.</p>http://cvgmt.sns.it/paper/4001/Shape derivatives for minima of integral functionalshttp://cvgmt.sns.it/paper/4000/G. Bouchitté, I. Fragalà, I. Lucardesi.<p>For $\Omega$ varying amongopen bounded sets in $\mathbb R^n$, we consider shape functionals $J (\Omega)$ defined as the infimum over a Sobolev space of an integral energy of the kind$\int _\Omega[ f (\nabla u) + g (u) ]$, under Dirichlet or Neumann conditions on $\partial \Omega$.Under fairly weak assumptions on the integrands $f$ and $g$, we prove that, when a given domain $\Omega$ is deformed into a one-parameter family of domains $\Omega _\varepsilon$ through an initial velocity field $V\in W ^ {1, \infty} (\mathbb R^n, \mathbb R^n)$, the corresponding shape derivative of $J$ at $\Omega$ in the direction of $V$ exists. Under some further regularity assumptions, we show that the shape derivative can be represented as a boundary integral depending linearly on the normal component of $V$ on $\partial \Omega$. Our approach to obtain the shape derivative is new, and it is based on the joint use of Convex Analysis and Gamma-convergence techniques. It allows to deduce, as a companion result, optimality conditions in the form of conservation laws.</p>http://cvgmt.sns.it/paper/4000/A nonstandard free boundary problem arising in the shape optimization of thin torsion rodshttp://cvgmt.sns.it/paper/3999/J. J. Alibert, G. Bouchitté, I. Fragalà, I. Lucardesi.<p>We study a 2d-variational problem, in which the cost functional is an integral depending onthe gradient through a convex but not strictly convex integrand, and the admissible functionshave zero gradient on the complement of a given domain D. We are interested in establishingwhether solutions exist whose gradient “avoids” the region of non-strict convexity. Actually, the answer to this question is related to establishing whether homogenization phenomena occur in optimal thin torsion rods. We provide some existence results for different geometries of D, andwe study the nonstandard free boundary problem with a gradient obstacle, which is obtainedthrough the optimality conditions.</p>http://cvgmt.sns.it/paper/3999/Optimal thin torsion rods and Cheeger setshttp://cvgmt.sns.it/paper/3998/G. Bouchitté, I. Fragalà, I. Lucardesi, P. Seppecher.<p>We carry out an asymptotic analysis of the following shape optimization problem: a given volume fraction of elastic material must be distributed in a cylindrical design region of infinitesimal cross section in order to maximize resistance to a twisting load. We derive a limit rod model written in different equivalent formulations and for which we are able to give necessary and sufficient conditions characterizing optimal configurations. Eventually we show that for a convex design region and for very small volume fractions, the optimal shape tends to concentrate section by section near the boundary of the Cheeger set of the design. These results were announced in <a href='G. Bouchitté, I. Fragalà, and P. Seppecher, C. R. Math., 348 (2010), pp. 467–471'>G. Bouchitté, I. Fragalà, and P. Seppecher, C. R. Math., 348 (2010), pp. 467–471</a>.</p>http://cvgmt.sns.it/paper/3998/Interpolations and Fractional Sobolev Spaces in Carnot Groupshttp://cvgmt.sns.it/paper/3997/A. Maalaoui, A. Pinamonti.<p>In this paper we present an interpolation approach to the fractional Sobolev spaces in Carnot groups using the K-method. This approach provides us with a different characterization of these Sobolev spaces, moreover, it provides us with the limiting behavior of the fractional Sobolev norms at the end-points. This allows us to deduce results similar to the Bourgain-Brezis-Mironescu and Maz'ya-Shaposhnikova in the case $p>1$ and D\'{a}vila's result in the case $p=1$. Also, this allows us to deduce the limiting behavior of the fractional perimeter in Carnot groups.</p>http://cvgmt.sns.it/paper/3997/A short story on optimal transport and its many applicationshttp://cvgmt.sns.it/paper/3996/F. Santambrogio.<p>We present some examples of optimal transport problems and of applications to different sciences (logistics, economics, image processing, and a little bit of evolution equations) through the crazy story of an industrial dynasty regularly asking advice from an exotic mathematician.</p><p>This is a popular science paper written for MFO after a conference on the applications of optimal transport. It is intended for general (scientific-oriented) audience.</p>http://cvgmt.sns.it/paper/3996/A homogenization result in the gradient theory of phase transitionshttp://cvgmt.sns.it/paper/3994/R. Cristoferi, I. Fonseca, A. Hagerty, C. Popovici.<p>A variational model in the context of the gradient theory for fluid-fluid phase transitions with small scale heterogeneities is studied.In particular, the case where the scale $\e$ of the small homogeneities is of the same order of the scale governing the phase transition is considered.The interaction between homogenization and the phase transitions process will lead, in the limit as $\e\to0$, to an anisotropic interfacial energy.</p>http://cvgmt.sns.it/paper/3994/Phase field systems with maximal monotone nonlinearities related to sliding mode control problemshttp://cvgmt.sns.it/paper/3993/M. Colturato.http://cvgmt.sns.it/paper/3993/Well-posedness and longtime behavior for a singular phase field system with perturbed phase dynamicshttp://cvgmt.sns.it/paper/3992/M. Colturato.<p>We consider a singular phase field system located in a smooth bounded domain. In the entropy balance equation appears a logarithmic nonlinearity. The second equation of the system, deduced from a balance law for the microscopic forces that are responsible for the phase transition process, is perturbed by an additional term involving a possibly nonlocal maximal monotone operator and arising from a class of sliding mode control problems. We prove existence and uniqueness of the solution for this resulting highly nonlinear system. Moreover, under further assumptions, the longtime behavior of the solution is investigated.</p>http://cvgmt.sns.it/paper/3992/Energy release rate and stress intensity factors in planar elasticity in presence of smooth crackshttp://cvgmt.sns.it/paper/3991/S. Almi, I. Lucardesi.<p>In this work we first analyze the singular behavior of the displacement $u$ of a linearly elastic body in dimension 2 close to the tip of a smooth crack, extending the well-known results for straight fractures to general smooth ones. As conjectured by Griffith, $u$ behaves as the sum of an $H^2$-function and a linear combination of two singular functions whose profile is similar to the square root of the distance from the tip. The coefficients of the linear combination are the so called stress intensity factors. Afterwards, we prove the differentiability of the elastic energy with respect to an infinitesimalfracture elongation and we compute the energy release rate, enlightening its dependence on the stressintensity factors.</p>http://cvgmt.sns.it/paper/3991/Energy-dissipation balance of a smooth moving crackhttp://cvgmt.sns.it/paper/3990/M. Caponi, I. Lucardesi, E. Tasso.<p>In this paper we provide necessary and sufficient conditions in order to guarantee the energy-dissipation balance of a Mode III crack, growing on a prescribed smooth path. Moreover, we characterize the singularity of the displacement near the crack tip, generalizing the result in <a href='S. Nicaise, A.M. Sandig - J. Math.Anal. Appl., 2007'>S. Nicaise, A.M. Sandig - J. Math.Anal. Appl., 2007</a> valid for straight fractures.</p><p>Preprint SISSA 31$/$2018$/$MATE</p>http://cvgmt.sns.it/paper/3990/GRADIENT ESTIMATES FOR PERTURBED ORNSTEIN-UHLENBECK SEMIGROUPS ON INFINITE DIMENSIONAL CONVEX DOMAINShttp://cvgmt.sns.it/paper/3989/L. Angiuli, S. Ferrari, D. Pallara.http://cvgmt.sns.it/paper/3989/Discrete-to-continuum limits of particles with an annihilation rulehttp://cvgmt.sns.it/paper/3988/M. Morandotti, P. van Meurs.<p>In the recent trend of extending discrete-to-continuum limit passages for gradient flows of single-species particle systems with singular and nonlocal interactions to particles of opposite sign, any annihilation effect of particles with opposite sign has been side-stepped. We present the first rigorous discrete-to-continuum limit passage which includes annihilation. This result paves the way to applications such as vortices, charged particles, and dislocations.In more detail, the discrete setting of our discrete-to-continuum limit passage is given by particles on the real line. Particles of the same type interact by a singular interaction kernel; those of opposite sign interact by a regular one. If two particles of opposite sign collide, they annihilate, i.e., they are taken out of the system. The challenge for proving a discrete-to-continuum limit is that annihilation is an intrinsically discrete effect where particles vanish instantaneously in time, while on the continuum scale the mass of the particle density decays continuously in time.The proof contains two novelties: (i) the empirical measures of the discrete dynamics (with annihilation rule) satisfy the continuum evolution equation that only implicitly encodes annihilation, and (ii) by imposing a relatively mild separation assumption on the initial data we can identify the limiting particle density as a solution to the same continuum evolution equation.</p>http://cvgmt.sns.it/paper/3988/A multi-material transport problem with arbitrary marginalshttp://cvgmt.sns.it/paper/3987/A. Marchese, A. Massaccesi, S. Stuvard, R. Tione.<p>In this paper we study general transportation problems in $\mathbb{R}^n$, in which $m$ different goods are moved simultaneously. The initial and final displacements of the goods are represented by measures $\mu^-$, $\mu^+$ on $\mathbb{R}^n$ with values in $\mathbb{R}^m$. When the measures are finite atomic, a discrete transportation network is a measure $T$ on $\mathbb{R}^n$ with values in $\mathbb{R}^{n\times m}$ represented by an oriented graph $\mathcal{G}$ in $\mathbb{R}^n$ whose edges carry multiplicities in $\mathbb{R}^m$. The constraint is encoded in the relation ${\rm div}(T)=\mu^--\mu^+$. The cost of the discrete transportation $T$ is obtained integrating on $\mathcal{G}$ a very general function $\mathcal{C}:\mathbb{R}^m\to\mathbb{R}$ of the multiplicity. The proof of the existence of minimizers for arbitrary (possibly diffuse) data $(\mu^-,\mu^+)$ requires an explicit formula for the relaxation, on arbitrary transportation networks, of the functional on graphs defined above. Under additional assumptions on $\mathcal{C}$, we prove the existence of transportation networks with finite cost and the stability of the minimizers with respect to variations of the given data. The proofs of the main results of the paper require notions from the theory of currents with coefficients in a group. In the process, we give details of the proof of a useful result concerning the relaxation of general functionals defined on polyhedral chains with coefficients in groups.</p>http://cvgmt.sns.it/paper/3987/Observability inequalities for transport equations through Carleman estimateshttp://cvgmt.sns.it/paper/3986/P. Cannarsa, G. Floridia, M. Yamamoto.<p> We consider the transport equation $\ppp_t u(x,t) + H(t)\cdot \nabla u(x,t) =0$ in $\OOO\times(0,T),$ where $T>0$ and $\OOO\subset \R^d $ is a boundeddomain with smooth boundary $\ppp\OOO$. First, we prove a Carleman estimate forsolutions of finite energy with piecewise continuous weight functions. Then,under a further condition on $H$ which guarantees that the orbit $\{H(t)\in\R^d, \thinspace 0 \le t \le T\}$intersects $\ppp\OOO$, we prove an energy estimate which inturn yields an observability inequality. Our results are motivated byapplications to inverse problems.</p>http://cvgmt.sns.it/paper/3986/Differential of metric valued Sobolev mapshttp://cvgmt.sns.it/paper/3985/N. Gigli, E. Pasqualetto, E. Soultanis.<p>We introduce a notion of differential of a Sobolev map between metric spaces. The differential is given in the framework of tangent and cotangent modules of metric measure spaces, developed by the first author. We prove that our notion is consistent with Kirchheim's metric differential when the source is a Euclidean space, and with the abstract differential provided by the first author when the target is $\mathbb{R}$.</p>http://cvgmt.sns.it/paper/3985/Bounded variation and relaxed curvature of surfaceshttp://cvgmt.sns.it/paper/3984/D. Mucci, A. Saracco.<p>We consider a relaxed notion of energy of non-parametric codimension one surfaces that takes account ofarea, mean curvature, and Gauss curvature. It is given by the best value obtained by approximation with inscribed polyhedral surfaces.The BV and measure properties of functions with finite relaxed energy are studied.Concerning the total mean and Gauss curvature, the classical counterexample by Schwarz-Peano to the definition of area is also analyzed.</p>http://cvgmt.sns.it/paper/3984/Existence of calibrations for the Mumford-Shah functional and the reinitialization of the distance function in the framework of Chan and Vese algorithmshttp://cvgmt.sns.it/paper/3983/M. Carioni.<p>Revised version</p>http://cvgmt.sns.it/paper/3983/