cvgmt Papershttp://cvgmt.sns.it/papers/en-usWed, 17 Oct 2018 11:36:22 +0000Four dimensional closed manifolds admit a weak harmonic Weyl metrichttp://cvgmt.sns.it/paper/4075/G. Catino, P. Mastrolia, D. D. Monticelli, F. Punzo.<p>On four-dimensional closed manifolds we introduce a class of canonical Riemannian metrics, that we call weak harmonic Weyl metrics, defined as critical points in the conformal class of a quadratic functional involving the norm of the divergence of the Weyl tensor. This class includes Einstein and, more in general, harmonic Weyl manifolds. We prove that every closed four-manifold admits a weak harmonic Weyl metric, which is the unique (up to dilations) minimizer of the functional in a suitable conformal class. In general the problem is degenerate elliptic due to possible vanishing of the Weyl tensor. In order to overcome this issue, we minimize the functional in the conformal class determined by a reference metric, constructed by Aubin, with nowhere vanishing Weyl tensor. Moreover, we show that anti-self-dual metrics with positive Yamabe invariant can be characterized by pinching conditions involving suitable quadratic Riemannian functionals.</p>http://cvgmt.sns.it/paper/4075/Finer estimates on the 2-dimensional matching problemhttp://cvgmt.sns.it/paper/4074/L. Ambrosio, F. Glaudo.<p>We study the asymptotic behaviour of the expected cost of the random matching problem on a 2-dimensional compact manifold, improving in several aspects the results of L. Ambrosio, F. Stra and D. Trevisan (A PDE approach to a 2-dimensional matching problem). In particular, we simplify the original proof (by treating at the same time upper and lower bounds) and we obtain the coefficient of the leading term of the asymptotic expansion of the expected cost for the random bipartite matching on a general 2-dimensional closed manifold. We also sharpen the estimate of the error term given by M. Ledoux (On optimal matching of Gaussian samples II) for the semi-discrete matching. </p><p>As a technical tool, we develop a refined contractivity estimate for the heat flow on random data that might be of independent interest.</p>http://cvgmt.sns.it/paper/4074/Free boundary regularity for a multiphase shape optimization problemhttp://cvgmt.sns.it/paper/4073/L. Spolaor, B. Trey, B. Velichkov.<p>In this paper we prove a $C^{1,\alpha}$ regularity result in dimension two for almost-minimizers of the constrained one-phase Alt-Caffarelli and the two-phase Alt-Caffarelli-Friedman functionals for an energy with variable coefficients. As a consequence, we deduce the complete regularity of solutions of a multiphase shape optimization problem for the first eigenvalue of the Dirichlet-Laplacian up to the fixed boundary. One of the main ingredient is a new application of the epiperimetric-inequality of Spolaor-Velichkov (CPAM, 2018) up to the boundary. While the framework that leads to this application is valid in every dimension, the epiperimetric inequality is known only in dimension two, thus the restriction on the dimension.</p>http://cvgmt.sns.it/paper/4073/Dirichlet conditions in Poincaré-Sobolev inequalities: the sub-homogeneous casehttp://cvgmt.sns.it/paper/4072/D. Zucco.<p>We investigate the dependence of optimal constants in Poincaré-Sobolev inequalities of planar domains on the region where the Dirichlet condition is imposed. More precisely, we look for the best Dirichlet regions, among closed and connected sets with prescribed total length L (one-dimensional Hausdorff measure), that make these constants as small as possible. We study their limiting behaviour, showing, in particular, that Dirichlet regions homogenize inside the domain with comb-shaped structures, periodically distribuited at different scales and with different orientations. To keep track of these information we rely on a Γ-convergence result in the class of varifolds. This also permits applications to reinforcements of anisotropic elastic membranes. At last, we provide some evidences for a conjecture.</p>http://cvgmt.sns.it/paper/4072/Two-well rigidity and multidimensional sharp-interface limits for solid-solid phase transitionshttp://cvgmt.sns.it/paper/4071/E. Davoli, M. Friedrich.<p>We establish a quantitative rigidity estimate for two-well frame-indifferent nonlinear energies, in the case in which the two wells have exactly one rank-one connection. Building upon this novel rigidity result, we then analyze solid-solid phase transitions in arbitrary space dimensions, under a suitable anisotropic penalization of second variations. By means of $\Gamma$-convergence, we show that, as the size of transition layers tends to zero, singularly perturbed two-well problems approach an effective sharp-interface model. The limiting energy is finite only for deformations which have the structure of a laminate. In this case, it is proportional to the total length of the interfaces between the two phases.</p>http://cvgmt.sns.it/paper/4071/Approximation of the relaxed perimeter functional under a connectedness constraint by phase-fieldshttp://cvgmt.sns.it/paper/4070/P. W. Dondl, M. Novaga, B. Wirth, S. Wojtowytsch.<p>We develop a phase-field approximation of the relaxation of the perimeter functional in the plane under a connectedness constraint based on the classical Modica-Mortola functional and the connectedness constraint of $[8]$. We prove convergence of the approximating energies and present numerical results and applications to image segmentation.</p><p>$[8]$ P. W. Dondl, A. Lemenant, and S. Wojtowytsch. Phase Field Models for Thin Elastic Structures with Topological Constraint. Arch. Ration. Mech. Anal., 223(2):693–736, 2017.</p>http://cvgmt.sns.it/paper/4070/Convergence and non-convergence of many-particle evolutions with multiple signshttp://cvgmt.sns.it/paper/4069/A. Garroni, Mark A. Peletier, L. Scardia, P. van Meurs.<p> We address the question of convergence of evolving interacting particlesystems as the number of particles tends to infinity. We consider two types ofparticles, called positive and negative. Same-sign particles repel each other,and opposite-sign particles attract each other. The interaction potential isthe same for all particles, up to the sign, and has a logarithmic singularityat zero. The central example of such systems is that of dislocations incrystals. Because of the singularity in the interaction potential, the discreteevolution leads to blow-up in finite time. We remedy this situation byregularising the interaction potential at a length-scale $\delta_n>0$, whichconverges to zero as the number of particles $n$ tends to infinity. We establish two main results. The first one is an evolutionary convergenceresult showing that the empirical measures of the positive and of the negativeparticles converge to a solution of a set of coupled PDEs which describe theevolution of their continuum densities. In the setting of dislocations thesePDEs are known as the Groma-Balogh equations. In the proof we rely on thetheory of $\lambda$-convex gradient flows, a priori estimates for theGroma-Balogh equations and Orlicz spaces. The proof require $\delta_n$ toconverge to zero sufficiently slowly. The second result is a counterexample,demonstrating that if $\delta_n$ converges to zero sufficiently fast, then thelimits of the empirical measures of the positive and the negative dislocationsdo not satisfy the Groma-Balogh equations. These results show how the validity of the Groma-Balogh equations as thelimit of many-particle systems depends in a subtle way on the scale at whichthe singularity of the potential is regularised.</p>http://cvgmt.sns.it/paper/4069/A convex approach to the Gilbert-Steiner problemhttp://cvgmt.sns.it/paper/4068/M. Bonafini, E. Oudet.<p>We describe a convex relaxation for the Gilbert-Steiner problem both in $R^d$ and on manifolds, extending the framework proposed in <a href='9'>9</a>, and we discuss its sharpness by means of calibration type arguments. The minimization of the resulting problem is then tackled numerically and we present results for an extensive set of examples. In particular we are able to address the Steiner tree problem on surfaces.</p>http://cvgmt.sns.it/paper/4068/Asymptotic behavior of BV functions and sets of finite perimeter in metric measure spaceshttp://cvgmt.sns.it/paper/4067/S. Eriksson-Bique, J. Gill, P. Lahti, N. Shanmugalingam.<p>In this paper, we study the asymptotic behavior of BV functions in complete metric measurespaces equipped with a doubling measure supporting a $1$-Poincar\'e inequality. We show that at almost every point $x$ outside the Cantor and jump parts of a BV function, the asymptotic limitof the function is a Lipschitz continuous function of least gradient on a tangent space to the metricspace based at $x$. We also show that, at co-dimension $1$ Hausdorff measurealmost every measure-theoretic boundary point of a set $E$ of finite perimeter, there is an asymptoticlimit set $(E)_\infty$ corresponding to the asymptotic expansion of $E$ and that every such asymptotic limit $(E)_\infty$ is a quasiminimal set of finite perimeter. We also show that the perimeter measure of $(E)_\infty$ is Ahlfors co-dimension $1$ regular.</p>http://cvgmt.sns.it/paper/4067/Critical weak-$L^{p}$ differentiability of singular integralshttp://cvgmt.sns.it/paper/4066/L. Ambrosio, A. Ponce, R. Rodiac.<p>We establish that for every function $u \in L^1_\mathrm{loc}(\Omega)$ whose distributional Laplacian $\Delta u$ is a signed Borel measure in an open set $\Omega$ in $\mathbb{R}^N$, the distributional gradient $\nabla u$ is differentiable almost everywhere in $\Omega$ with respect to the weak-$L^{\frac{N}{N-1}}$ Marcinkiewicz norm.We show in addition that the absolutely continuous part of $\Delta u$ with respect to the Lebesgue measure equals zero almost everywhere on the level sets $\{u = \alpha\}$ and $\{\nabla u =e\}$, for every $\alpha \in \mathbb{R}$ and $e \in \mathbb{R}^N$. Our proofs rely on an adaptation of Calderón and Zygmund's singular-integral estimates inspired by subsequent work by Hajlasz.</p>http://cvgmt.sns.it/paper/4066/Multiphase Free Discontinuity Problems: Monotonicity Formula and Regularity Resultshttp://cvgmt.sns.it/paper/4065/D. Bucur, I. Fragalà, A. Giacomini.<p>The purpose of this paper is to analyze regularity properties of local solutionsto free discontinuity problems characterized by the presence of multiple phases. Local solutions are meant according to an ad hoc, nonstandard notion of {\it multiphase local almost-quasi minimizers}. In particular this notion penalizes, among contacts between two different phases, only those which occur at jump points, leaving for free no-jump interfaces which may occur at the zero level of the corresponding state functions. In this setting, our main result states that the phases are open and the jump set (globally considered for all the phases) is essentially closed and Ahlfors regular. This is the same kind of regularity holding for one phase local almost-quasi minimizers of general free discontinuity problems. To achieve the same target in presence of multiple phases demands to set up new refined tools. They are a multiphase monotonicity formulaand a multiphase decay lemma, which extend respectively the corresponding one phase results by Bucur-Luckhaus and De Giorgi-Carriero-Leaci. The proof of the former relies on a sharp collective Sobolev extension result for functions with disjoint supports on a sphere, which may be of independent interest.</p>http://cvgmt.sns.it/paper/4065/Local minimality results for the Mumford-Shah functional via monotonicityhttp://cvgmt.sns.it/paper/4064/D. Bucur, I. Fragalà, A. Giacomini.<p>Let $\Omega\subseteq R^2$ be a bounded piecewise $C^{1,1}$ open set with convex corners, and let$MS(u):=\int_\Omega \vert \nabla u\vert^2\,dx+\alpha {\mathcal H}^1(J_u)+\beta\int_\Omega(u-g)^2 dx$be the Mumford-Shah functional on the space $SBV(\Omega)$, where $g\in L^\infty(\Omega)$ and $\alpha,\beta>0$. We prove that the function $u\in H^1(\Omega)$ such that$$\begin{cases}-\Delta u+\beta u=\beta g&\text{in }\Omega\\\frac{\partial u}{\partial \nu}=0 &\text{on }\partial\Omega\end{cases}$$is a local minimizer of $MS$ with respect to the $L^1$-topology. This is obtained as an application of interior and boundary monotonicity formulas for a weak notion of quasi minimizers of the Mumford-Shah energy. The local minimality result is then extended to more general free discontinuity problems taking into account also boundary conditions.</p>http://cvgmt.sns.it/paper/4064/Sign changing solutions of Poisson's equationhttp://cvgmt.sns.it/paper/4063/D. Bucur, M. van den Berg.<p> Let $\Omega$ be an open, possibly unbounded, set in Euclidean space $\R^m$,let $A$ be a measurable subset of $\Omega$ with measure $<br>A<br>$, and let $\gamma\in (0,1)$. We investigate whether the solution $v_{\Om,A,\gamma}$ of $-\Deltav=\gamma{\bf 1}_{\Omega-A}-(1-\gamma){\bf 1}_{A},\, v\in H_0^1(\Omega)$ changessign. Bounds are obtained for $<br>A<br>$ in terms of geometric characteristics of$\Om$ (bottom of the spectrum of the Dirichlet Laplacian, torsion, measure, or$R$-smoothness of the boundary) such that ${\rm essinf} v_{\Om,A,\gamma}\ge 0$.We show that ${\rm essinf} v_{\Om,A,\gamma}<0$ for any measurable set $A$,provided $<br>A<br> >\gamma <br>\Om<br>$. This value is sharp. We also study the shapeoptimisation problem of the optimal location of $A$ (with prescribed measure)which minimises the essential infimum of $v_{\Om,A,\gamma}$. Surprisingly, if$\Om$ is a ball, a symmetry breaking phenomenon occurs.</p>http://cvgmt.sns.it/paper/4063/Geometric control of the Robin Laplacian eigenvalues: the case of negative boundary parameterhttp://cvgmt.sns.it/paper/4062/D. Bucur, S. Cito.<p>This paper is motivated by the study of the existence of optimal domains maximizing the $k$-th Robin Laplacian eigenvalue among sets of prescribed measure, in the case of a negative boundary parameter. We answer positively to this question and prove an existence result in the class of measurable sets and for quite general spectral functionals. The key tools of our analysis rely on tight isodiametric and isoperimetric geometric controls of the eigenvalues. In two dimensions of the space, under simply connectedness assumptions, further qualitative properties are obtained on the optimal sets.</p>http://cvgmt.sns.it/paper/4062/Weighted Beckmann problem with boundary costshttp://cvgmt.sns.it/paper/4061/S. Dweik.<p>We show that a solution to a variant of the Beckmann problem can be obtained by studying the limit of some weighted p-Laplacian problems. In addition, we connect this problem to a formulation with Kantorovich potentials with Dirichlet boundary conditions.</p>http://cvgmt.sns.it/paper/4061/Higher order Gamma-limits for singularly perturbed Dirichlet-Neumann problemshttp://cvgmt.sns.it/paper/4060/G. Gravina, G. Leoni.<p> A mixed Dirichlet-Neumann problem is regularized with a family of singularlyperturbed Neumann-Robin boundary problems, parametrized by $\varepsilon > 0$.Using an asymptotic development by Gamma-convergence, the asymptotic behaviorof the solutions to the perturbed problems is studied as $\varepsilon \to 0^+$,recovering classical results in the literature.</p>http://cvgmt.sns.it/paper/4060/Dimension reduction problems in the modelling of hydrogel thin filmshttp://cvgmt.sns.it/paper/4059/D. Lučić.http://cvgmt.sns.it/paper/4059/Dimension reduction for thin films with transversally varying prestrain: the oscillatory and the non-oscillatory casehttp://cvgmt.sns.it/paper/4058/M. Lewicka, D. Lučić.<p>We study the non-Euclidean (incompatible) elastic energy functionals in thedescription of prestressed thin films, at their singular limits ($\Gamma$-limits)as $h\to 0$ in the film's thickness $h$. Firstly, we extend the priorresults <a href='Lewicka-Pakzad, Bhattacharya-Lewicka-Schaffner, Lewicka-Raoult-Ricciotti'>Lewicka-Pakzad, Bhattacharya-Lewicka-Schaffner, Lewicka-Raoult-Ricciotti</a> to arbitrary incompatibility metricsthat depend on both the midplate and the transversal variables (the``non-oscillatory'' case). Secondly, we analyze a more general class ofincompatibilities, where the transversal dependence of the lower orderterms is not necessarily linear (the ``oscillatory'' case), extendingthe results of <a href='Agostiniani-Lucic-Lucantonio, Schmidt'>Agostiniani-Lucic-Lucantonio, Schmidt</a> to arbitrary metrics and higher order scalings. We exhibitconnections between the two cases via projections of appropriatecurvature forms on the polynomial tensor spaces. We also show theeffective energy quantisation in terms of scalings as a power of $h$and discuss the scaling regimes $h^2$ (Kirchhoff), $h^4$ (vonKarman), $h^6$ in the general case, and all possible (even power)regimes for conformal metrics. Thirdly, we prove thecoercivity inequalities for the singular limits at $h^2$- and $h^4$-scaling orders, while disproving the full coercivity of the classical von Karman energyfunctional at scaling $h^4$.</p>http://cvgmt.sns.it/paper/4058/Structural and geometric properties of $\sf RCD$ spaceshttp://cvgmt.sns.it/paper/4057/E. Pasqualetto.http://cvgmt.sns.it/paper/4057/Derivation of a heteroepitaxial thin-film modelhttp://cvgmt.sns.it/paper/4056/E. Davoli, P. Piovano.<p>A variational model for epitaxially-strained thin films on rigid substrates is derived both by {\Gamma}-convergence from a transition-layer setting, and by relaxation from a sharp-interface description available in the literature for regular configurations. The model is characterized by a configurational energy that accounts for both the competing mechanisms responsible for the film shape. On the one hand, the lattice mismatch between the film and the substrate generate large stresses, and corrugations may be present because film atoms move to release the elastic energy. On the other hand, flatter profiles may be preferable to minimize the surface energy. Some first regularity results are presented for energetically-optimal film profiles.</p>http://cvgmt.sns.it/paper/4056/