CVGMT Papershttp://cvgmt.sns.it/papers/en-usThu, 26 Apr 2018 01:51:33 +0000Weyl scalars on compact Ricci solitonshttp://cvgmt.sns.it/paper/3860/G. Catino, P. Mastrolia.<p>We investigate the triviality of compact Ricci solitons under general scalar conditions involving the Weyl tensor. More precisely, we show that a compact Ricci soliton is Einstein if a generic linear combination of divergences of the Weyl tensor contracted with suitable covariant derivatives of the potential function vanishes. In particular we recover and improve all known related results. This paper can be thought as a first, preliminary step in a general program which aims at showing that Ricci solitons can be classified finding a ``generic'' $[k, s]$-vanishing condition on the Weyl tensor, for every $k, s\in\mathbb{N}$, where $k$ is the order of the covariant derivatives of Weyl and $s$ is the type of the (covariant) tensor involved.</p>http://cvgmt.sns.it/paper/3860/On the Cauchy problem for the wave equation on time-dependent domainshttp://cvgmt.sns.it/paper/3859/G. Dal Maso, R. Toader.<p>We introduce a notion of solution to the wave equation on a suitable class of time-dependent domains and compare it with a previous definition. We prove an existence result for the solution of the Cauchy problem and present some additional conditions which imply uniqueness.</p>http://cvgmt.sns.it/paper/3859/A note on homogeneous Sobolev spaces of fractional orderhttp://cvgmt.sns.it/paper/3858/L. Brasco, A. Salort.<p>We consider a homogeneous fractional Sobolev space obtained by completion of the space of smooth test functions, with respect to a Sobolev--Slobodecki\u{\i} norm. We compare it to the fractional Sobolev space obtained by the $K-$method in real interpolation theory. We show that the two spaces do not always coincide and give some sufficient conditions on the open sets for this to happen. We also highlight some unnatural behaviors of the interpolation space. The treatment is as self-contained as possible.</p>http://cvgmt.sns.it/paper/3858/Optimal one-dimensional reinforcement for elastic membraneshttp://cvgmt.sns.it/paper/3857/G. Alberti, G. Buttazzo, S. Guarino Lo Bianco, E. Oudet.<p>In this paper we study the optimal reinforcement of an elastic membrane, fixed at its boundary, by means of a connected one-dimensional structure. We show the existence of an optimal solution that may present multiplicities, that is regions where the optimal structure overlaps. Some numerical simulations are shown to confirm this issue and to illustrate the complexity of the optimal structures when their total length becomes large.</p>http://cvgmt.sns.it/paper/3857/Regularity of the free boundary for the vectorial Bernoulli problemhttp://cvgmt.sns.it/paper/3856/D. Mazzoleni, S. Terracini, B. Velichkov.<p>In this paper we study the regularity of the free boundary for a vector-valued Bernoulli problem, with no sign assumptions on the boundary data. More precisely, given an open, smooth set of finite measure $D\subset \mathbb{R}^d$, $\Lambda>0$ and $\varphi_i\in H^{1/2}(\partial D)$, we consider the free boundary problem \[\min{\Big\{\sum_{i=1}^k\int_D\vert\nabla v_i\vert^2+\Lambda\,\mathcal L^d\left(\bigcup_{i=1}^k\{v_i\not=0\}\right)\;:\;v_i=\varphi_i\;on \;\partial D\Big\}}.\]We prove that, for any optimal vector $U=(u_1,\dots, u_k)$, the free boundary $\partial (\cup_{i=1}^k\{u_i\not=0\})\cap D$ is made by a regular part, which is relatively open and locally the graph of a $C^\infty$ function, a (one-phase) singular part, of Hausdorff dimension at most $d-d^*$, for a $d^*\in\{5,6,7\}$, and by a set of branching (two-phase) points, which is relatively closed and of finite $(d-1)$-dimensional Hausdorff measure.Our arguments are based on the NTA structure of the regular part of the free boundary.</p>http://cvgmt.sns.it/paper/3856/Prescribing Gaussian and geodesic curvature on the diskhttp://cvgmt.sns.it/paper/3855/S. Cruz Blázquez, D. Ruiz.<p>In this paper we consider the problem of prescribing theGaussian and geodesic curvature on a disk and its boundary, respectively,via a conformal change of the metric. This leads us to a Liouville-typeequation with a nonlinear Neumann boundary condition. We addressthe question of existence by setting the problem in a variational frameworkwhich seems to be completely new in the literature. We are able tofind minimizers under symmetry assumptions.</p>http://cvgmt.sns.it/paper/3855/Constancy of the dimension for RCD(K,N) spaces via regularity of Lagrangian flowshttp://cvgmt.sns.it/paper/3854/E. Bruè, D. Semola.<p>We prove a regularity result for Lagrangian flows of Sobolev vector fields over RCD(K,N)metric measure spaces, regularity is understood with respect to a newly defined quasi-metricbuilt from the Green function of the Laplacian. Its main application is that RCD(K,N) spaceshave constant dimension. In this way we generalize to such abstract framework a result provedby Colding-Naber for Ricci limit spaces, introducing ingredients that are new even in thesmooth setting.</p>http://cvgmt.sns.it/paper/3854/Non-degeneracy and uniqueness of solutions to singular mean field equations on bounded domainshttp://cvgmt.sns.it/paper/3853/D. Bartolucci, A. Jevnikar, C. S. Lin.<p>The aim of this paper is to complete the program initiated in [50], [23] and then carried out by several authors concerning non-degeneracy and uniqueness of solutionsto mean field equations. In particular, we consider mean field equations with general singular data on non-smooth domains. The argument is based on the Alexandrov-Bol inequality and on the eigenvalues analysis of linearized singular Liouville-type problems.</p>http://cvgmt.sns.it/paper/3853/Best constants for two families of higher order critical Sobolev embeddingshttp://cvgmt.sns.it/paper/3852/I. Shafrir, D. Spector.<p>In this paper we obtain the best constants in some higher order Sobolev inequalities in the critical exponent. These inequalities can be separated into two types: those that embed into $L^\infty(\mathbb{R}^N)$ and those that embed into slightly larger target spaces. Concerning the former, we show that for $k \in \{1,\ldots, N-1\}$, $N-k$ even, one has an optimal constant $c_k>0$ such that\[ \vert\vert u \vert\vert_{L^\infty} \leq c_k \int \vert \nabla^k (-\Delta)^{(N-k)/2} u\vert\]for all $u \in C^\infty_c(\mathbb{R}^N)$ (the case $k=N$ was handled in previous work of the first author). Meanwhile the most significant of the latter is a variation of D. Adams' higher order inequality of J. Moser: For $\Omega \subset \mathbb{R}^N$, $m \in \mathbb{N}$ and $p=\frac{N}{m}$, there exists $A>0$ and optimal constant $\beta_0>0$ such that\[ \int_{\Omega} \exp (\beta_0 \vert u \vert^{p^\prime}) \leq A \vert \Omega \vert \]for all $u$ such that $\vert\vert \nabla^m u\vert\vert_{L^p(\Omega)} \leq 1$, where $\vert\vert \nabla^m u\vert\vert_{L^p(\Omega)}$ is the traditional semi-norm on the space $W^{m,p}(\Omega)$.</p>http://cvgmt.sns.it/paper/3852/Uniqueness of Equilibrium with Sufficiently Small Strains in Finite Elasticityhttp://cvgmt.sns.it/paper/3851/D. Spector, S. Spector.<p>The uniqueness of equilibrium for a compressible,hyperelastic body subject to dead-load boundary conditions isconsidered. It is shown, for both the displacement andmixed problems,that there cannot be two solutions of the equilibriumequations of Finite (Nonlinear) Elasticity whose nonlinear strainsare uniformly close to each other. This result is analogous tothe result of Fritz John (Comm.\ Pure Appl.\ Math.\ \textbf{25},617--634, 1972) who proved that, for the displacement problem,there is a most one equilibrium solution with uniformly small strains.The proof in this manuscript utilizes Geometric Rigidity; a newstraightforward extensionof the Fefferman-Stein inequality to bounded domains;and, an appropriate adaptation, for Elasticity,of a result from the Calculus of Variations.Specifically, it is herein shown thatthe uniform positivity of the second variation of the energy at an equilibrium solution implies that this mapping isa local minimizer of the energy among deformations whose gradient is sufficiently close, in $BMO\cap\, L^1$, to the gradient of the equilibrium solution.</p>http://cvgmt.sns.it/paper/3851/Minimal cluster computation for four planar regions with the same areahttp://cvgmt.sns.it/paper/3850/E. Paolini, A. Tamagnini.<p>The topology of a minimal cluster of four planar regions with equal areas and smallest possibleperimeter was found in <a href='http://cvgmt.sns.it/paper/3120/'>(9)</a>. Here we describe the computation used to check that the symmetric cluster withthe given topology is indeed the unique minimal cluster.</p>http://cvgmt.sns.it/paper/3850/Minimal-Time Mean Field Gameshttp://cvgmt.sns.it/paper/3849/G. Mazanti, F. Santambrogio.<p>This paper considers a mean field game model inspired by crowd motion where agents want to leave a given bounded domain through a part of its boundary in minimal time. Each agent is free to move in any direction, but their maximal speed is bounded in terms of the average density of agents around their position in order to take into account congestion phenomena.</p><p>After a preliminary study of the corresponding minimal-time optimal control problem, we formulate the mean field game in a Lagrangian setting and prove existence of Lagrangian equilibria using a fixed point strategy. We provide a further study of equilibria under the assumption that agents may leave the domain through the whole boundary, in which case equilibria are described through a system of a continuity equation on the distribution of agents coupled with a Hamilton--Jacobi equation on the value function of the optimal control problem solved by each agent. This is possible thanks to the semiconcavity of the value function, which follows from some further regularity properties of optimal trajectories obtained through Pontryagin Maximum Principle. Simulations illustrate the behavior of equilibria in some particular situations.</p>http://cvgmt.sns.it/paper/3849/Quantitative estimates for regular Lagrangian flows with $BV$ vector fieldshttp://cvgmt.sns.it/paper/3848/Q. H. Nguyen.<p>In this paper, we solve an open problem mentioned in \cite{AmbCrip}. Exactly, we prove the well posedness of regular Lagrangian flows to vector fields $\mathbf{B}=(\mathbf{B}^1,...,\mathbf{B}^d)\in L^1((0,T);L^1\cap L^\infty(\mathbb{R}^d))$ satisfying $\mathbf{B}^i=\sum_{j=1}^{m}\mathbf{K}_j^i*b_j,$ $b_j\in L^1((0,T),BV(\mathbb{R}^d))$and $\operatorname{div}(\mathbf{B})\in L^1((0,T);L^\infty(\mathbb{R}^d))$ for $d\geq 2$, where $(\mathbf{K}_j^i)_{i,j}$ are singular kernels in $\mathbb{R}^d$.</p>http://cvgmt.sns.it/paper/3848/How a minimal surface leaves a thin obstaclehttp://cvgmt.sns.it/paper/3846/M. Focardi, E. Spadaro.<p>We prove optimal regularity and a detailed analysis of the free boundary of the solutions to the thin obstacle problem for nonparametric minimal surfaces with flat obstacles.</p>http://cvgmt.sns.it/paper/3846/Asymptotic limit of linear parabolic equations with spatio-temporal degenerated potentialshttp://cvgmt.sns.it/paper/3845/P. Álvarez-Caudevilla, M. Bonnivard, A. Lemenant.<p>In this paper, we observe how the heat equation in a non-cylindrical domain can arise as the asymptotic limit of a parabolic problem in a cylindrical domain, by adding a potential that vanishes outside the limit domain. This can be seen as a parabolic version of a previous work by the first and last authors, concerning the stationary case~\cite{PaAn}. We provide a strong convergence result for the solution by use of energetic methods and $\Gamma$-convergence technics. Then, we establish an exponential decay estimate coming from an adaptation of an argument by B. Simon.</p>http://cvgmt.sns.it/paper/3845/Local uniqueness of $m$-bubbling sequences for the Gel'fand equationhttp://cvgmt.sns.it/paper/3844/D. Bartolucci, A. Jevnikar, Y. Lee, W. Yang.<p>We consider the Gel'fand problem, $\Delta w_{\varepsilon}+\varepsilon^2he^{w_{\varepsilon}}=0$ in $\Omega$, $w_{\varepsilon}=0$ on $\partial\Omega,$where $h$ is a nonnegative function in ${\Omega\subset\mathbb{R}^2}$. Under suitable assumptions on $h$ and $\Omega$, we prove the local uniqueness of $m-$bubbling solutions for any $\varepsilon>0$ small enough.</p>http://cvgmt.sns.it/paper/3844/Sharp boundary behaviour of solutions to semilinear nonlocal elliptic equationshttp://cvgmt.sns.it/paper/3843/M. Bonforte, A. Figalli, J. L. Vazquez.<p>We investigate quantitative properties of nonnegative solutions $u(x)\ge 0$ to the semilinear diffusion equation $\mathcal L u= f(u)$, posed in a bounded domain $\Omega\subset \mathbb R^N$ with appropriate homogeneous Dirichlet or outer boundary conditions. The operator $\mathcal L$ may belong to a quite general class of linear operators that include the standard Laplacian, the two most common definitions of the fractional Laplacian $(-\Delta)^s$ ($0<s<1$) in a bounded domain with zero Dirichlet conditions, and a number of other nonlocal versions. The nonlinearity $f$ is increasing and looks like a power function $f(u)\sim u^p$, with $p\le 1$.</p><p>The aim of this paper is to show sharp quantitative boundary estimates based on a new iteration process. We also prove that, in the interior, solutions are H\"older continuous and even classical (when the operator allows for it). In addition, we get H\"older continuity up to the boundary.</p><p>Particularly interesting is the behaviour of solution when the number $\frac{2s}{1-p}$ goes below the exponent $\gamma \in(0,1]$ corresponding tothe H\"older regularity of the first eigenfunction $\mathcal L\Phi_1=\lambda_1 \Phi_1$.Indeed a change of boundary regularity happens in the different regimes $\frac{2s}{1-p} \gtreqqless \gamma$,and in particular a logarithmic correction appears in the ``critical'' case $\frac{2s}{1-p} = \gamma$.</p><p>For instance, in the case of the spectral fractional Laplacian, this surprising boundary behaviour appears in the range $0<s\leq (1-p)/2$.</p>http://cvgmt.sns.it/paper/3843/On the isoperimetric problem with double densityhttp://cvgmt.sns.it/paper/3842/A. Pratelli, G. Saracco.<p> In this paper we consider the isoperimetric problem with double density in anEuclidean space, that is, we study the minimisation of the perimeter amongsubsets of $\mathbb{R}^n$ with fixed volume, where volume and perimeter arerelative to two different densities. The case of a single density, orequivalently, when the two densities coincide, has been well studied in thelast years; nonetheless, the problem with two different densities is animportant generalisation, also in view of applications. We will prove theexistence of isoperimetric sets in this context, extending the known resultsfor the case of single density.</p>http://cvgmt.sns.it/paper/3842/Anisotropic tubular neighborhoods in euclidean spaceshttp://cvgmt.sns.it/paper/3841/A. Chambolle, L. Lussardi, E. Villa.<p>Let $E \subset \mathbb R^N$ be a compact set and $C\subset \mathbb R^N$ be a convex body with $0\in{\rm int}\,C$. We prove that the topological boundary of the anisotropic enlargement $E+rC$ is contained in a finite union of Lipschitz surfaces and we investigate the regularity of the volume function $V_E(r):=<br>E+rC<br>$ proving that up to a countable set $V_E$ is of class $C^1$.</p>http://cvgmt.sns.it/paper/3841/Soap film spanning electrically repulsive elastic protein linkshttp://cvgmt.sns.it/paper/3840/G. Bevilacqua, L. Lussardi, A. Marzocchi.<p>We study the equilibrium problem of a mechanical system consisting by two Kirchhoff rods linked in an arbitrary way and also forming knots, constrained not to touch themselves by means of electrical repulsion and tied by a soap film, as a model to describe the interaction between an electrically charged protein and a biomembrane. We prove the existence of a solution with minimum total energy, which may be quite irregular, by using techniques of the Calculus of Variations.</p>http://cvgmt.sns.it/paper/3840/