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We prove (sub)mean value formulas at the point $0\in\Sigma$ for (sub)harmonic functions a on a hypersurface $\Sigma\subset\mathbb{R}^{n+1}$ where the differentiable structure and the surface measure depend on the ambient Grushin structure.

We consider a class of infinite weighted metric trees obtained as perturbations of self-similar regular trees. Possible definitions of the boundary traces of functions in the Sobolev space on such a structure are discussed by using identifications of the tree boundary with a surface. Our approach unifies some constructions proposed by Maury, Salort, Vannier (2009) for dyadic discrete weighted trees (expansion in orthogonal bases of harmonic functions on the graph and using Haar-type bases on the domain representing the boundary), and by Nicaise, Semin (2018) and Joly, Kachanovska, Semin (2019) for fractal metric trees (approximation by finite sections and identification of the boundary with a interval): we show that both machineries give the same trace map, and for a range of parameters we establish the precise Sobolev regularity of the traces. In addition, we introduce new geometric ingredients by proposing an identification with arbitrary Riemannian manifolds. It is shown that any compact manifold admits a suitable multiscale decomposition and, therefore, can be identified with a metric tree boundary in the context of trace theorems., Comment: 67 pages. Several wrong citations were corrected

We consider non-negative $\sigma$-finite measure spaces coupled with a proper functional $P$ that plays the role of a perimeter. We introduce the Cheeger problem in this framework and extend many classical results on the Cheeger constant and on Cheeger sets to this setting, requiring minimal assumptions on the pair measure space-perimeter. Throughout the paper, the measure space will never be asked to be metric, at most topological, and this requires the introduction of a suitable notion of Sobolev spaces, induced by the coarea formula with the given perimeter., Comment: 52 pages

In this paper we introduce a notion of magnetic field in the Heisenberg group and we study its influence on spectral properties of the corresponding magnetic (sub-elliptic) Laplacian. We show that uniform magnetic fields uplift the bottom of the spectrum. For magnetic fields vanishing at infinity, including Aharonov--Bohm potentials, we derive magnetic improvements to a variety of Hardy-type inequalities for the Heisenberg sub-Laplacian. In particular, we establish a sub-Riemannian analogue of Laptev and Weidl sub-criticality result for magnetic Laplacians in the plane. Instrumental for our argument is the validity of a Hardy-type inequality for the Folland--Stein operator, that we prove in this paper and has an interest on its own., Comment: 44 pages

We say that a control system is locally controllable if the attainable set from any state $x$ contains an open neighborhood of $x$, while it is controllable if the attainable set from any state is the entire state manifold. We show in this note that a control system satisfying local controllability is controllable. Our self-contained proof is alternative to the combination of two previous results by Kevin Grasse., Comment: 9 pages, 4 figures, 2 tables; this updated version includes new citations to previous literature

We consider the isoperimetric problem for clusters in the plane with a double density, that is, perimeter and volume depend on two weights. In this paper we consider the isotropic case, in the parallel paper "On the Steiner property for planar minimizing clusters. The anisotropic case", the anisotropic case is studied. Here we prove that, in a wide generality, minimal clusters enjoy the "Steiner property", which means that the boundaries are made by ${\rm C}^{1,\gamma}$ regular arcs, meeting in finitely many triple points with the $120^\circ$ property., Comment: Scheme of the proof and references added. arXiv admin note: text overlap with arXiv:2106.08099

In this paper we discuss the Steiner property for minimal clusters in the plane with an anisotropic double density. This means that we consider the classical isoperimetric problem for clusters, but volume and perimeter are defined by using two densities. In particular, the perimeter density may also depend on the direction of the normal vector. The classical "Steiner property" for the Euclidean case (which corresponds to both densities being equal to $1$) says that minimal clusters are made by finitely many ${\rm C}^{1,\gamma}$ arcs, meeting in finitely many "triple points". We can show that this property holds under very weak assumptions on the densities. In the parallel paper "On the Steiner property for planar minimizing clusters. The isotropic case" we consider the isotropic case, i.e., when the perimeter density does not depend on the direction, which makes most of the construction much simpler. In particular, in the present case the three arcs at triple points do not necessarily meet with three angles of $120^\circ$, which is instead what happens in the isotropic case., Comment: Scheme of the proof and references added

We consider Wilson-Cowan-type models for the mathematical description of orientation-dependent Poggendorff-like illusions. Our modelling improves two previously proposed cortical-inspired approaches embedding the sub-Riemannian heat kernel into the neuronal interaction term, in agreement with the intrinsically anisotropic functional architecture of V1 based on both local and lateral connections. For the numerical realisation of both models, we consider standard gradient descent algorithms combined with Fourier-based approaches for the efficient computation of the sub-Laplacian evolution. Our numerical results show that the use of the sub-Riemannian kernel allows to reproduce numerically visual misperceptions and inpainting-type biases in a stronger way in comparison with the previous approaches.

We study the isoperimetric problem for anisotropic left-invariant perimeter measures on $\mathbb R^3$, endowed with the Heisenberg group structure. The perimeter is associated with a left-invariant norm $\phi$ on the horizontal distribution. We first prove a representation formula for the $\phi$-perimeter of regular sets and, assuming some regularity on $\phi$ and on its dual norm $\phi^*$, we deduce a foliation property by sub-Finsler geodesics of $\mathrm C^2$-smooth surfaces with constant $\phi$-curvature. We then prove that the characteristic set of $\mathrm C^2$-smooth surfaces that are locally extremal for the isoperimetric problem is made of isolated points and horizontal curves satisfying a suitable differential equation. Based on such a characterization, we characterize $\mathrm C^2$-smooth $\phi$-isoperimetric sets as the sub-Finsler analogue of Pansu's bubbles. We also show, under suitable regularity properties on $\phi$, that such sub-Finsler candidate isoperimetric sets are indeed $\mathrm C^2$-smooth. By an approximation procedure, we finally prove a conditional minimality property for the candidate solutions in the general case (including the case where $\phi$ is crystalline).

We consider the evolution model proposed in [9, 6] to describe illusory contrast perception phenomena induced by surrounding orientations. Firstly, we highlight its analogies and differences with the widely used Wilson-Cowan equations [48], mainly in terms of efficient representation properties. Then, in order to explicitly encode local directional information, we exploit the model of the primary visual cortex (V1) proposed in [20] and largely used over the last years for several image processing problems [24,38,28]. The resulting model is thus defined in the space of positions and orientation and it is capable to describe assimilation and contrast visual bias at the same time. We report several numerical tests showing the ability of the model to reproduce, in particular, orientation-dependent phenomena such as grating induction and a modified version of the Poggendorff illusion. For this latter example, we empirically show the existence of a set of threshold parameters differentiating from inpainting to perception-type reconstructions and describing long-range connectivity between different hypercolumns in V1., Comment: This is the revised extended invited journal version of the SSVM 2019 conference proceeding arXiv:1812.07425

In this work we have aimed to reproduce supra-threshold perception phenomena, specifically visual illusions, with Wilson-Cowan-type models of neuronal dynamics. We have found that it is indeed possible to do so, but that the ability to replicate visual illusions is related to how well the neural activity equations comply with the efficient representation principle. Our first contribution is to show that the Wilson-Cowan equations can reproduce a number of brightness and orientation-dependent illusions, and that the latter type of illusions require that the neuronal dynamics equations consider explicitly the orientation, as expected. Then, we formally prove that there can't be an energy functional that the Wilson-Cowan equations are minimizing, but that a slight modification makes them variational and yields a model that is consistent with the efficient representation principle. Finally, we show that this new model provides a better reproduction of visual illusions than the original Wilson-Cowan formulation.

In this paper we study various Hardy inequalities in the Heisenberg group $\mathbb H^n$, w.r.t. the Carnot-Carath\'eodory distance $\delta$ from the origin. We firstly show that the optimal constant for the Hardy inequality is strictly smaller than $n^2 = (Q-2)^2/4$, where $Q$ is the homogenous dimension. Then, we prove that, independently of $n$, the Heisenberg group does not support a radial Hardy inequality, i.e., a Hardy inequality where the gradient term is replaced by its projection along $\nabla_{\mathbb H}\delta$. This is in stark contrast with the Euclidean case, where the radial Hardy inequality is equivalent to the standard one, and has the same constant. Motivated by these results, we consider Hardy inequalities for non-radial directions, i.e., directions tangent to the Carnot-Carath\'eodory balls. In particular, we show that the associated constant is bounded on homogeneous cones $C_\Sigma$ with base $\Sigma\subset \mathbb S^{2n}$, even when $\Sigma$ degenerates to a point. This is a genuinely sub-Riemannian behavior, as such constant is well-known to explode for homogeneous cones in the Euclidean space.

In this paper we show that, for a sub-Laplacian $\Delta$ on a $3$-dimensional manifold $M$, no point interaction centered at a point $q_0\in M$ exists. When $M$ is complete w.r.t. the associated sub-Riemannian structure, this means that $\Delta$ acting on $C^\infty_0(M\setminus\{q_0\})$ is essentially self-adjoint. A particular example is the standard sub-Laplacian on the Heisenberg group. This is in stark contrast with what happens in a Riemannian manifold $N$, whose associated Laplace-Beltrami operator is never essentially self-adjoint on $C^\infty_0(N\setminus\{q_0\})$, if $\dim N\le 3$. We then apply this result to the Schr\"odinger evolution of a thin molecule, i.e., with a vanishing moment of inertia, rotating around its center of mass.

We consider a differential model describing neuro-physiological contrast perception phenomena induced by surrounding orientations. The mathematical formulation relies on a cortical-inspired modelling [10] largely used over the last years to describe neuron interactions in the primary visual cortex (V1) and applied to several image processing problems [12,19,13]. Our model connects to Wilson-Cowan-type equations [23] and it is analogous to the one used in [3,2,14] to describe assimilation and contrast phenomena, the main novelty being its explicit dependence on local image orientation. To confirm the validity of the model, we report some numerical tests showing its ability to explain orientation-dependent phenomena (such as grating induction) and geometric-optical illusions [21,16] classically explained only by filtering-based techniques [6,18].

We address the double bubble problem for the anisotropic Grushin perimeter $P_\alpha$, $\alpha\geq 0$, and the Lebesgue measure in $\mathbb R^2$, in the case of two equal volumes. We assume that the contact interface between the bubbles lays on either the vertical or the horizontal axis. Since no regularity theory is available in this setting, in both cases we first prove existence of minimizers via the direct method by symmetrization arguments and then characterize them in terms of the given area by first variation techniques. Angles at which minimal boundaries intersect satisfy the standard 120-degree rule up to a suitable change of coordinates. While for $\alpha=0$ the Grushin perimeter reduces to the Euclidean one and both minimizers coincide with the symmetric double bubble found in [Foisy et al., Pacific J. Math. (1993)], for $\alpha=1$ vertical interface minimizers have Grushin perimeter strictly greater than horizontal interface minimizers. As the latter ones are obtained by translating and dilating the Grushin isoperimetric set found in [Monti Morbidelli, J. Geom. Anal. (2004)], we conjecture that they solve the double bubble problem with no assumptions on the contact interface.

We prove a general essential self-adjointness criterion for sub-Laplacians on complete sub-Riemannian manifolds, defined with respect to singular measures. As a consequence, we show that the intrinsic sub-Laplacian (i.e. defined w.r.t. Popp's measure) is essentially self-adjoint on the equiregular connected components of a sub-Riemannian manifold. This result holds under mild regularity assumptions on the singular region, and when the latter does not contain characteristic points., Comment: 21 pages, 1 figure

We study a family of spheres with constant mean curvature (CMC) in the Riemannian Heisenberg group $H^1$. These spheres are conjectured to be the isoperimetric sets of $H^1$. We prove several results supporting this conjecture. We also focus our attention on the sub-Riemannian limit.

We study a variational problem for the perimeter associated with the Grushin plane, called minimal partition problem with trace constraint. This consists in studying how to enclose three prescribed areas in the Grushin plane, using the least amount of perimeter, under an additional "one-dimensional" constraint on the intersections of their boundaries. We prove existence of regular solutions for this problem, and we characterize them in terms of isoperimetric sets, showing differences with the Euclidean case. The problem arises from the study of quantitative isoperimetric inequalities and has connections with the theory of minimal clusters., Comment: 20 pages, 7 figures

In the Heisenberg group H^n, we prove quantitative isoperimetric inequalities for Pansu's spheres, that are known to be isoperimetric under various assumptions. The inequalities are shown for suitably restricted classes of competing sets and the proof relies on the construction of sub-calibrations.

We study the isoperimetric problem in H-type groups and Grushin spaces, emphasizing a relation between them. We prove existence, symmetry and regularity properties of isoperimetric sets, under a symmetry assumption that depends on the dimension.