cvgmt Papershttps://cvgmt.sns.it/papers/en-usSun, 25 Sep 2022 08:22:15 +0000Uniform $C^{1,\alpha}$-regularity for almost-minimizers of some nonlocal perturbations of the perimeterhttps://cvgmt.sns.it/paper/5722/M. Goldman, B. Merlet, M. Pegon.<p>In this paper, we establish a $C^{1,\alpha}$-regularity theorem for almost-minimizers of the functional $\mathcal{F}_{\varepsilon,\gamma}=P-\gamma P_{\varepsilon}$, where $\gamma\in(0,1)$ and $P_{\varepsilon}$ is a nonlocal energy converging to the perimeter as $\varepsilon$ vanishes.<br>Our theorem provides a criterion for $C^{1,\alpha}$-regularity at a point of the boundary which is <i>uniform</i> as the parameter $\varepsilon$ goes to $0$.<br>As a consequence we obtain that volume-constrained minimizers of $\mathcal{F}_{\varepsilon,\gamma}$ are balls for any $\varepsilon$ small enough. For small $\varepsilon$, this minimization problem corresponds to the large mass regime for a Gamow-type problem where the nonlocal repulsive term is given by an integrable kernel $G$ with sufficiently fast decay at infinity.</p>https://cvgmt.sns.it/paper/5722/Two rigidity results for stable minimal hypersurfaceshttps://cvgmt.sns.it/paper/5721/G. Catino, P. Mastrolia, A. Roncoroni.<p>The aim of this paper is to prove two results concerning the rigidity of complete, immersed, orientable, stable minimal hypersurfaces: we show that they are hyperplane in $\mathbb{R}^4$, while they do not exist in positively curved closed Riemannian $(n+1)$-manifold when $n\leq 5$; in particular, there are no stable minimal hypersurfaces in $\mathbb{S}^{n+1}$ when $n\leq 5$. The first result was recently proved also by Chodosh and Li, and the second is a consequence of a more general result concerning minimal surfaces with finite index. Both theorems rely on a conformal method, inspired by a classical work of Fischer-Colbrie.</p>https://cvgmt.sns.it/paper/5721/Intrinsic sub-Laplacian for hypersurface in a contact sub-Riemannian manifoldhttps://cvgmt.sns.it/paper/5720/D. Barilari, K. Habermann.<p>We construct and study the intrinsic sub-Laplacian, defined outside the set of characteristic points, for a smooth hypersurface embedded in a contact sub-Riemannian manifold. We prove that, away from characteristic points, the intrinsic sub-Laplacian arises as the limit of Laplace--Beltrami operators built by means of Riemannian approximations to the sub-Riemannian structure using the Reeb vector field. We carefully analyse three families of model cases for this setting obtained by considering canonical hypersurfaces embedded in model spaces for contact sub-Riemannian manifolds. In these model cases, we show that the intrinsic sub-Laplacian is stochastically complete and in particular, that the stochastic process induced by the intrinsic sub-Laplacian almost surely does not hit characteristic points.</p>https://cvgmt.sns.it/paper/5720/Rigidity estimates for isometric and conformal maps from $\mathbb{S}^{n-1}$ to $\mathbb{R}^n$https://cvgmt.sns.it/paper/5719/S. Luckhaus, K. Zemas.<p>We investigate both linear and nonlinear stability aspects of rigid motions (resp. Möbius transformations) of $\mathbb{S}^{n-1}$ among Sobolev maps from $\mathbb{S}^{n-1}$ into $\mathbb{R}^n$. Unlike similar in flavour results for maps defined on domains of $\mathbb{R}^n$ and mapping into $\mathbb{R}^n$, not only an isometric (resp. conformal) deficit is necessary in this more flexible setting, but also a deficit measuring the distortion of $\mathbb{S}^{n-1}$ under the maps in consideration. The latter is defined as an associated isoperimetric type of deficit. The focus is mostly on the case $n=3$ (where it is explained why the estimates are optimal in their corresponding settings), but we also address the necessary adaptations for the results in higher dimensions. We also obtain linear stability estimates for both cases in all dimensions. These can be regarded as Korn-type inequalities for the combination of the quadratic form associated with the isometric (resp. conformal) deficit on $\mathbb{S}^{n-1}$ and the isoperimetric one.</p>https://cvgmt.sns.it/paper/5719/A note on a rigidity estimate for degree $\pm 1$ conformal maps on $\mathbb{S}^2$https://cvgmt.sns.it/paper/5718/J. Hirsch, K. Zemas.<p>In this note, we present a short alternative proof of an estimate obtained by Mantel, Muratov and Simon in (Arch Rational Mech. Anal. 239 (2021), 219–299) regarding the rigidity of degree $\pm 1$ conformal maps of $\mathbb{S}^2$, that is, its Möbius transformations.</p>https://cvgmt.sns.it/paper/5718/Stability of the vortex in micromagnetics and related modelshttps://cvgmt.sns.it/paper/5717/X. Lamy, E. Marconi.<p>We consider line-energy models of Ginzburg-Landau type in a two-dimensional simply connectedbounded domain. Congurations of vanishing energy have been characterizedby Jabin, Otto and Perthame: the domain must be a disk, and the conguration a vortex.We prove a quantitative version of this statement in the class of $C^{1,1}$ domains, improvingon previous results by Lorent. In particular, the deviation of the domain from a diskis controlled by a power of the energy, and that power is optimal. The main tool is aLagrangian representation introduced by the second author, which allows to decomposethe energy along characteristic curves.</p>https://cvgmt.sns.it/paper/5717/Structural changes in nonlocal denoising models arising through bi-level parameter learninghttps://cvgmt.sns.it/paper/5716/E. Davoli, R. Ferreira, C. Kreisbeck, H. Schönberger.<p>We introduce a unified framework based on bi-level optimization schemes to deal with parameter learning in the context of image processing. The goal is to identify the optimal regularizer within a family depending on a parameter in a general topological space. Our focus lies on the situation with non-compact parameter domains, which is, for example, relevant when the commonly used box constraints are disposed of. To overcome this lack of compactness, we propose a natural extension of the upper-level functional to the closure of the parameter domain via Gamma-convergence, which captures possible structural changes in the reconstruction model at the edge of the domain. Under two main assumptions, namely, Mosco-convergence of the regularizers and uniqueness of minimizers of the lower-level problem, we prove that the extension coincides with the relaxation, thus admitting minimizers that relate to the parameter optimization problem of interest. We apply our abstract framework to investigate a quartet of practically relevant models in image denoising, all featuring nonlocality. The associated families of regularizers exhibit qualitatively different parameter dependence, describing a weight factor, an amount of nonlocality, an integrability exponent, and a fractional order, respectively. After the asymptotic analysis that determines the relaxation in each of the four settings, we finally establish theoretical conditions on the data that guarantee structural stability of the models and give examples of when stability is lost.</p>https://cvgmt.sns.it/paper/5716/Multi-material model and shape optimization for bending and torsion of inextensible rodshttps://cvgmt.sns.it/paper/5715/P. W. Dondl, A. Maione, S. Wolff-Vorbeck.<p>We derive a model for the optimization of bending and torsional rigidity of non-homogeneous elastic rods, by studying a sharp interface shape optimization problem with perimeter penalization for the rod cross section, that treats the resulting torsional and bending rigidities as objectives. We then formulate a phase field approximation to the optimization problem and show Γ-convergence to the aforementioned sharp interface model. This also implies existence of minimizers for the sharp interface optimization problem. Finally, we numerically find minimizers of the phase field problem using a steepest descent approach and relate the resulting optimal shapes to the development of plant morphology.</p>https://cvgmt.sns.it/paper/5715/BV estimates on the transport density with Dirichlet region on the boundaryhttps://cvgmt.sns.it/paper/5714/S. Dweik.<p>In this paper, we prove BV regularity on the transport density in the mass transport problem to the boundary in two dimension under certain conditions on the domain, the boundary cost and the mass distribution. Moreover, we show by a counter-example that the smoothness of the mass distribution, the boundary and the boundary cost does not imply that the transport density is W<sup>{1,p},</sup> for some p > 1.</p>https://cvgmt.sns.it/paper/5714/Gluing non-unique Navier-Stokes solutionshttps://cvgmt.sns.it/paper/5713/D. Albritton, E. Bruè, M. Colombo.<p>We construct non-unique Leray solutions of the forced Navier-Stokes equations in bounded domains via gluing methods. This demonstrates a certain locality and robustness of the non-uniqueness discovered by the authors in <a href='1'>1</a>.</p>https://cvgmt.sns.it/paper/5713/A note on the supersolution method for Hardy's inequalityhttps://cvgmt.sns.it/paper/5712/F. Bianchi, L. Brasco, F. Sk, A. C. Zagati.<p>We prove a characterization of Hardy's inequality in Sobolev-Slobodeckii spaces in terms of positive local weak supersolutions of the relevant Euler-Lagrange equation. This extends previous results by Ancona and Kinnunen & Korte for standard Sobolev spaces. The proof is based on variational methods.</p>https://cvgmt.sns.it/paper/5712/On the sharp Hardy inequality in Sobolev-Slobodeckii spaceshttps://cvgmt.sns.it/paper/5711/F. Bianchi, L. Brasco, A. C. Zagati.<p>We study the sharp constant in the Hardy inequality for fractional Sobolev spaces defined on open subsets of the Euclidean space. We first list some properties of such a constant, as well as of the associated variational problem. We then restrict the discussion to open convex sets and compute such a sharp constant, by constructing suitable supersolutions by means of the distance function. Such a method of proof works only for $s\,p\ge 1$ or for $\Omega$ being a half-space. We exhibit a simple example suggesting that this method can not work for $s\,p<1$ and $\Omega$ different from a half-space. The case $s\,p<1$ for a generic convex set is left as an interesting open problem, except in the Hilbertian setting (i.e. for $p=2$): in this case we can compute the sharp constant in the whole range $0<s<1$. This completes a result which was left open in the literature.</p>https://cvgmt.sns.it/paper/5711/Density of subalgebras of Lipschitz functions in metric Sobolev spaces and applications to Wasserstein Sobolev spaceshttps://cvgmt.sns.it/paper/5710/M. Fornasier, G. Savaré, G. E. Sodini.<p>We prove a general criterion for the density in energy of suitable subalgebras of Lipschitz functions in the metric-Sobolev space $H^{1,p}(X,\mathsf{d},\mathfrak{m})$ associated with a positive and finite Borel measure $\mathfrak{m}$ in a separable and complete metric space $(X,\mathsf{d})$.We then provide a relevant application to the case of the algebra of cylinder functions in the Wasserstein Sobolev space $H^{1,2}(\mathcal{P}_2(\mathbb{M}),W_{2},\mathfrak{m})$ arising from a positive and finite Borel measure $\mathfrak{m}$ on the Kantorovich-Rubinstein-Wasserstein space $(\mathcal{P}_2(\mathbb{M}),W_{2})$ of probability measures in a finite dimensional Euclidean space, a complete Riemannian manifold, or a separable Hilbert space $\mathbb{M}$. We will show that such a Sobolev space is always Hilbertian, independently of the choice of the reference measure $\mathfrak{m}$ so that the resulting Cheeger energy is a Dirichlet form.We will eventually provide an explicit characterization for the corresponding notion of $\mathfrak{m}$-Wasserstein gradient, showing useful calculus rules and its consistency with the tangent bundle and the $\Gamma$-calculus inherited from the Dirichlet form.</p>https://cvgmt.sns.it/paper/5710/Subgraphs of $\rm BV$ functions on $\rm RCD$ spaceshttps://cvgmt.sns.it/paper/5709/G. Antonelli, C. Brena, E. Pasqualetto.<p> In this work we extend classical results for subgraphs of functions ofbounded variation in $\mathbb{R}^n\times\mathbb{R}$ to the setting of$\mathsf{X}\times\mathbb{R}$, where $\mathsf{X}$ is an ${\rm RCD}(K,N)$ metricmeasure space. In particular, we give the precise expression of the push-forward onto$\mathsf{X}$ of the perimeter measure of the subgraph in$\mathsf{X}\times\mathbb{R}$ of a $\rm BV$ function on $\mathsf{X}$. Moreover,in properly chosen good coordinates, we write the precise expression of thenormal to the boundary of the subgraph of a $\rm BV$ function $f$ with respectto the polar vector of $f$, and we prove change-of-variable formulas.</p>https://cvgmt.sns.it/paper/5709/Optimal transport with nonlinear mobilities: a deterministic particle approximation resulthttps://cvgmt.sns.it/paper/5708/S. Di Marino, L. Portinale, E. Radici.<p> We study the discretization of generalized Wasserstein distances withnonlinear mobilities on the real line via suitable discrete metrics on the coneof N ordered particles, a setting which naturally appears in the framework ofdeterministic particle approximation of partial differential equations. Inparticular, we provide a $\Gamma$-convergence result for the associateddiscrete metrics as $N \to \infty$ to the continuous one and discussapplications to the approximation of one-dimensional conservation laws (ofgradient flow type) via the so-called generalized minimizing movements, provinga convergence result of the schemes at any given discrete time step $\tau>0$.This the first work of a series aimed at shedding new lights on the interplaybetween generalized gradient-flow structures, conservation laws, andWasserstein distances with nonlinear mobilities.</p>https://cvgmt.sns.it/paper/5708/A saturation phenomenon for a nonlinear nonlocal eigenvalue problemhttps://cvgmt.sns.it/paper/5707/F. Della Pietra, G. Piscitelli.<p> Given $1\le q \le 2$ and $\alpha\in\mathbb R$, we study the properties of thesolutions of the minimum problem \[\lambda(\alpha,q)=\min\left\{\dfrac{\displaystyle\int_{-1}^{1}<br>u'<br>^{2}dx+\alpha\left<br>\int_{-1}^{1}<br>u<br>^{q-1}u\,dx\right<br>^{\frac2q}}{\displaystyle\int_{-1}^{1}<br>u<br>^{2}dx}, u\inH_{0}^{1}(-1,1),\,u\not\equiv 0\right\}. \] In particular, depending on$\alpha$ and $q$, we show that the minimizers have constant sign up to acritical value of $\alpha=\alpha_{q}$, and when $\alpha>\alpha_{q}$ theminimizers are odd.</p>https://cvgmt.sns.it/paper/5707/On the second Dirichlet eigenvalue of some nonlinear anisotropic elliptic operatorshttps://cvgmt.sns.it/paper/5706/F. Della Pietra, N. Gavitone, G. Piscitelli.<p> Let $\Omega$ be a bounded open set of $\mathbb R^{n}$, $n\ge 2$. In thispaper we mainly study some properties of the second Dirichlet eigenvalue$\lambda_{2}(p,\Omega)$ of the anisotropic $p$-Laplacian \[ -\mathcalQ_{p}u:=-\textrm{div} \left(F^{p-1}(\nabla u)F_\xi (\nabla u)\right), \] where$F$ is a suitable smooth norm of $\mathbb R^{n}$ and $p\in]1,+\infty[$. Weprovide a lower bound of $\lambda_{2}(p,\Omega)$ among bounded open sets ofgiven measure, showing the validity of a Hong-Krahn-Szego type inequality.Furthermore, we investigate the limit problem as $p\to+\infty$.</p>https://cvgmt.sns.it/paper/5706/The anisotropic $\infty$-Laplacian eigenvalue problem with Neumann boundary conditionshttps://cvgmt.sns.it/paper/5704/G. Piscitelli.<p> We analize the limit problem of the anisotropic $p$-Laplacian as$p\rightarrow\infty$ with the mean of the viscosity solution. We also provesome geometric properties of eigenvalues and eigenfunctions. In particular, weshow the validity of a Szeg\"o-Weinberger type inequality.</p>https://cvgmt.sns.it/paper/5704/A sharp weighted anisotropic Poincaré inequality for convex domainshttps://cvgmt.sns.it/paper/5705/F. Della Pietra, N. Gavitone, G. Piscitelli.<p> We prove an optimal lower bound for the best constant in a class of weightedanisotropic Poincar\'e inequalities</p>https://cvgmt.sns.it/paper/5705/Saturation phenomena for some classes of nonlinear nonlocal eigenvalue problemshttps://cvgmt.sns.it/paper/5703/F. Della Pietra, G. Piscitelli.<p> Let us consider the following minimum problem \[ \lambda_\alpha(p,r)=\min_{\substack{u\in W_{0}^{1,p}(-1,1)\\u\not\equiv0}}\dfrac{\displaystyle\int_{-1}^{1}<br>u'<br>^{p}dx+\alpha\left<br>\int_{-1}^{1}<br>u<br>^{r-1}u\,dx\right<br>^{\frac pr}}{\displaystyle\int_{-1}^{1}<br>u<br>^{p}dx}, \] where$\alpha\in\mathbb R$, $p\ge 2$ and $\frac p2 \le r \le p$. We show that thereexists a critical value $\alpha_C=\alpha_C (p,r)$ such that the minimizers haveconstant sign up to $\alpha=\alpha_{C}$ and then they are odd when$\alpha>\alpha_{C}$.</p>https://cvgmt.sns.it/paper/5703/