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Given a bounded open connected set Ω ⊂ ℝ 2 {\Omega\subset\mathbb{R}^{2}} with Lipschitz boundary, we consider the class of piecewise constant maps u taking three fixed values α , β , γ ∈ ℝ 2 {\alpha,\beta,\gamma\in\mathbb{R}^{2}} , vertices of an equilateral triangle; for any u in this class, using a weak notion of Jacobian determinant valid for BV functions, we give a precise description of Det ⁡ ( ∇ ⁡ u ) {\operatorname{Det}(\nabla\kern 0.569055ptu)} and show that the relaxed graph area of u is bounded from above by a quantity related to the flat norm of Det ⁡ ( ∇ ⁡ u ) {\operatorname{Det}(\nabla\kern 0.569055ptu)} . The provided upper bound allows to show the validity of a De Giorgi conjecture regarding the relaxed area functional when one restricts to this class of piecewise constant functions. [ABSTRACT FROM AUTHOR]

We compute the relaxed Cartesian area for a general 0-homogeneous map of bounded variation, with respect to the strict BV-convergence. In particular, we show that the relaxed area is finite for this class of maps and we provide an integral representation formula. [ABSTRACT FROM AUTHOR]

We consider a gauge-invariant Ginzburg–Landau functional (also known as Abelian Yang–Mills–Higgs model), on Hermitian line bundles over closed Riemannian manifolds of dimension n ≥ 3 {n\geq 3} . Assuming a logarithmic energy bound in the coupling parameter, we study the asymptotic behaviour of critical points in the London limit. After a convenient choice of the gauge, we show compactness of finite-energy critical points in Sobolev norms. Moreover, thanks to a suitable monotonicity formula, we prove that the energy densities of critical points, rescaled by the logarithm of the coupling parameter, converge to the weight measure of a stationary, rectifiable varifold of codimension 2. [ABSTRACT FROM AUTHOR]

The Fletcher-Reeves (FR) method is widely recognized for its drawbacks, such as generating unfavorable directions and taking small steps, which can lead to subsequent poor directions and steps. To address this issue, we propose a modification to the FR method, and then we develop it into the three-term conjugate gradient method in this paper. The suggested methods, named "HZF" and "THZF", preserve the descent property of the FR method while mitigating the drawbacks. The algorithms incorporate strong Wolfe line search conditions to ensure effective convergence. Through numerical comparisons with other conjugate gradient algorithms, our modified approach demonstrates superior performance. The results highlight the improved efficacy of the HZF algorithm compared to the FR and three-term FR conjugate gradient methods. The new algorithm was applied to the problem of image restoration and proved to be highly effective in image restoration compared to other algorithms. [ABSTRACT FROM AUTHOR]

In this research, we adopt a comprehensive approach to address the multifractal and fractal analysis problem. We introduce a novel definition for the general Hausdorff and packing measures by considering sums involving certain functions and variables. Specifically, we explore the sums of the form ∑ i h - 1 (q h (μ (B (x i , r i))) + t g (r i)) , where μ represents a Borel probability measure on R d , and q and t are real numbers. The functions h and g are predetermined and play a significant role in our proposed intrinsic definition. Our observation reveals that estimating Hausdorff and packing pre-measures is significantly easier than estimating the exact Hausdorff and packing measures. Therefore, it is natural and essential to explore the relationships between the Hausdorff and packing pre-measures and their corresponding measures. This investigation constitutes the primary objective of this paper. Additionally, the secondary aim is to establish that, in the case of finite pre-measures, they possess a form of outer regularity in a metric space X that is not limited to a specific context or framework. [ABSTRACT FROM AUTHOR]

We study a variational model for soap films in which the films are represented by sets with fixed small volume rather than surfaces. In this problem, a minimizing sequence of completely “wet" films, or sets of finite perimeter spanning a wire frame, may converge to a film containing both wet regions of positive volume and collapsed (dry) surfaces. When collapsing occurs, these limiting objects lie outside the original minimization class and instead are admissible for a relaxed problem. Here we show that the relaxation and the original formulation are equivalent by approximating the collapsed films in the relaxed class by wet films in the original class. [ABSTRACT FROM AUTHOR]

For any Λ > 0 , let M n , Λ denote the space containing all locally Lipschitz minimal graphs of dimension n and of arbitrary codimension m in Euclidean space R n + m with uniformly bounded 2-dilation Λ of their graphic functions. In this paper, we show that this is a natural class to extend structural results known for codimension one. In particular, we prove that any tangent cone C of M ∈ M n , Λ at infinity has multiplicity one. This enables us to get a Neumann–Poincaré inequality on stationary indecomposable components of C. A corollary is a Liouville theorem for M. For small Λ > 1 (we can take any Λ < 2 ), we prove that (i) for n ≤ 7 , M is flat; (ii) for n > 8 and a non-flat M, any tangent cone of M at infinity is a multiplicity one quasi-cylindrical minimal cone in R n + m whose singular set has dimension ≤ n - 7 . [ABSTRACT FROM AUTHOR]

In this paper, we prove the isoperimetric inequality for the anisotropic Gaussian measure and characterize the cases of equality. We also find an example that shows Ehrhard symmetrization fails to decrease for the anisotropic Gaussian perimeter and gives a new inequality that includes an error term. This new inequality, in particular, gives us a hint to prove a uniqueness result for the anisotropic Ehrhard symmetrization. [ABSTRACT FROM AUTHOR]

We give sufficient conditions such that a volume preserving 1-Lipschitz map from a metric integral current onto an infinitesimally Euclidean Lipschitz manifold is an isometry. [ABSTRACT FROM AUTHOR]

In this work, we extend classical results for subgraphs of functions of bounded variation in R n × R to the setting of X × R , where X is an RCD (K , N) metric measure space. In particular, we give the precise expression of the push-forward onto X of the perimeter measure of the subgraph in X × R of a BV function on X . Moreover, in properly chosen good coordinates, we write the precise expression of the normal to the boundary of the subgraph of a BV function f with respect to the polar vector of f, and we prove change-of-variable formulas. [ABSTRACT FROM AUTHOR]

Working with sets of infinite Ψ t -Hausdorff measures can be cumbersome, so simplifying them to sets of positive finite general fractal measures proves to be highly beneficial. This paper aims to demonstrate that the Ψ t -Hausdorff measures adhere to the property of being a subset of positive and finite measures. Our main result is then applied to establish that the general fractal function dimension is defined as the supremum of the general fractal dimension across all Borel probability measures. [ABSTRACT FROM AUTHOR]

This paper establishes the global-in-time existence of a multi-phase mean curvature flow, evolving from an arbitrary closed rectifiable initial datum, which is a Brakke flow and a BV solution at the same time. In particular, we prove the validity of an explicit identity concerning the change of volume of the evolving grains, showing that their boundaries move according to the generalized mean curvature vector of the Brakke flow. As a consequence of the results recently established in [J. Fischer, S. Hensel, T. Laux and T. M. Simon, The local structure of the energy landscape in multiphase mean curvature flow: Weak-strong uniqueness and stability of evolutions, preprint 2020, https://arxiv.org/abs/2003.05478], under suitable assumptions on the initial datum, such additional property resolves the non-uniqueness issue of Brakke flows. [ABSTRACT FROM AUTHOR]

In this article, we prove the various minimality of the product of a 1-codimensional calibrated manifold and a paired calibrated set. This is motivated by the attempt to classify all possible singularities for Almgren minimal sets – Plateau's problem in the setting of sets. The Almgren minimality was introduced by Almgren to modernize Plateau's problem. It gives a very good description of local behavior for soap films. The natural question of whether the product of any two Almgren minimal sets is still minimal is still open, although it seems obvious in intuition. We prove the Almgren minimality for the product of two large classes of Almgren minimal sets – the class of 1-codimensional calibrated manifolds and the class of paired calibrated sets. The general idea is to properly combine different topological conditions (separation and spanning) under different homology groups, to set up a reasonable topological condition and prove the minimality for the product under this condition, which will imply the Almgren minimality. A main difficulty comes from the codimension – algebraic coherences such as multiplicity, separation and orientation do not exist anymore for codimensions larger than 1. An unexpectedly useful thing in the present paper is the flow of the calibrations. Its most important role among all is helping us to do the decomposition of a competitor with the help of the first projections along the flows. [ABSTRACT FROM AUTHOR]

In this paper, we study the regularity of several notions of Lipschitz solutions to the minimal surface system with an emphasis on partial regularity results. These include stationary solutions, integral weak solutions, and viscosity solutions. We also prove an interior gradient estimate for classical solutions to the system using the maximum principle, assuming the area-decreasing condition and that all but one component have small L ∞ norm. [ABSTRACT FROM AUTHOR]

The aim of this work is to discuss the proportionality of the multifractal measures. We will prove that the ratio of the multifractal measures is bounded. In addition, for a class of homogeneous Cantor sets, we find an explicit formula for their multifractal Hausdorff and packing function dimensions and discuss some interesting examples. [ABSTRACT FROM AUTHOR]

Nonparametric g-surfaces in Euclidean space have recently been characterized by Bildhauer–Fuchs in terms of closure of a 1-form associated to the so called asymptotic normal. This 1-form can be written by means of the pull-back of a canonical vector-valued 1-form through a suitable map depending on the asymptotic normal, that in the minimal surfaces case agrees with the Gauss graph map. We show that a similar characterization holds true for g-hypersurfaces of any high dimension N, but this time in terms of a canonical vector valued form of degree N - 1 . In the minimal hypersurfaces case, we finally discuss the lack of a relationship between the previous result and existence of good parameterizations, when N is greater than two. [ABSTRACT FROM AUTHOR]

The coarea formula is of crucial importance in analysis and has numerous applications in a variety of deep problems. Here we give a modern proof and discuss some of the applications. [ABSTRACT FROM AUTHOR]

We provide a fine description of the weak limit of sequences of regular axisymmetric maps with equibounded neo-Hookean energy, under the assumption that they have finite surface energy. We prove that these weak limits have a dipole structure, showing that the singular map described by Conti and De Lellis is generic in some sense. On this map, we provide the explicit relaxation of the neo-Hookean energy. We also make a link with Cartesian currents showing that the candidate for the relaxation we obtained presents strong similarities with the relaxed energy in the context of $\mathbb {S}^2$ -valued harmonic maps.

We prove that in arbitrary Carnot groups G of step 2, with a splitting G = W ⋅ L with L one-dimensional, the intrinsic graph of a continuous function φ : U ⊆ W → L is C H 1 -regular precisely when φ satisfies, in the distributional sense, a Burgers' type system Dφφ = ω, with a continuous ω. We stress that this equivalence does not hold already in the easiest step-3 Carnot group, namely the Engel group. We notice that our results generalize previous works by Ambrosio-Serra Cassano-Vittone and Bigolin-Serra Cassano in the setting of Heisenberg groups. As a tool for the proof we show that a continuous distributional solution φ to a Burgers' type system Dφφ = ω, with ω continuous, is actually a broad solution to Dφφ = ω. As a by-product of independent interest we obtain that all the continuous distributional solutions to Dφφ = ω, with ω continuous, enjoy 1/2-little Hölder regularity along vertical directions. [ABSTRACT FROM AUTHOR]

We establish the second variation of sub-Riemannian surface measure for minimal non-horizontal submanifolds of a sub-Riemannian stratified Lie group. We obtain some applications for codimension one. Furthermore, we present a new proof of the fact that the hyperbolic paraboloid is stable in the Heisenberg group. [ABSTRACT FROM AUTHOR]

The aim of this paper is to show that if the Hewitt–Stromberg pre-measures with respect to the gauge are finite, then these pre-measures have a kind of outer regularity in a general metric space X. We give also some conditions on the Hewitt–Stromberg pre-measures with respect to the gauge such that the Hewitt–Stromberg measures have an almost inner regularity on a complete separable metric space X. [ABSTRACT FROM AUTHOR]

We study local and global properties of solutions of the boundary value problem for the Dirac wave operator. This includes explicit formulas, domain of dependence, range of influence, finite propagation speed and energy estimates. We also introduce the space of Dirac wave homogeneous polynomials and the corresponding Dirac-wave conjugate functions. [ABSTRACT FROM AUTHOR]

Let H α = Δ - (α - 1) | x | α be a α -Hermite operator defined in R d with α ∈ [ 1 , ∞) . In this paper, we investigate subgraphs of α -Hermite BV functions and as a sample application, we deduce the rank-one theorem for the derivatives of vector-valued maps with H α -restricted BV functions. [ABSTRACT FROM AUTHOR]

We formulate the weak separation condition and the finite type condition for conformal iterated function systems on Riemannian manifolds with nonnegative Ricci curvature, and generalize the main theorems by Lau et al. (Monatsch Math 156:325–355, 2009). We also obtain a formula for the Hausdorff dimension of a self-similar set defined by an iterated function system satisfying the finite type condition, generalizing a corresponding result by Jin and Yau (Commun Anal Geom 13:821–843, 2005) and Lau and Ngai (Adv Math 208:647–671, 2007) on Euclidean spaces. Moreover, we obtain a formula for the Hausdorff dimension of a graph self-similar set generated by a graph-directed iterated function system satisfying the graph finite type condition, extending a result by Ngai et al. (Nonlinearity 23:2333–2350, 2010). [ABSTRACT FROM AUTHOR]

This paper introduces a notion of decompositions of integral varifolds into countably many integral varifolds, and the existence of such decomposition of integral varifolds whose first variation is representable by integration is established. However, the decompositions may fail to be unique. Furthermore, this result can be generalized by replacing the class of integral varifolds with some classes of rectifiable varifolds whose density is uniformly bounded from below; for these classes, we also prove a general version of the compactness theorem for integral varifolds. [ABSTRACT FROM AUTHOR]

In this paper, we attain the multifractal Hewitt–Stromberg dimension functions of Moran measures associated with homogeneous Moran fractals and show that the multifractal Hewitt–Stromberg measures are mutually singular for which the multifractal functions do not necessarily coincide. In particular, we give a positive answer to questions posed in Attia and Selmi (J Geom Anal 31:825-862, 2021) and discuss some interesting examples. [ABSTRACT FROM AUTHOR]

We obtain a comparison formula for integrals of mean curvatures of Riemannian hypersurfaces via Reilly’s identities. As applications, we derive several geometric inequalities for a convex hypersurface Γ\Gamma in a Cartan-Hadamard manifold MM. In particular, we show that the first mean curvature integral of a convex hypersurface γ\gamma nested inside Γ\Gamma cannot exceed that of Γ\Gamma , which leads to a sharp lower bound for the total first mean curvature of Γ\Gamma in terms of the volume it bounds in MM in dimension 3. This monotonicity property is extended to all mean curvature integrals when γ\gamma is parallel to Γ\Gamma , or MM has constant curvature. We also characterize hyperbolic balls as minimizers of the mean curvature integrals among balls with equal radii in Cartan-Hadamard manifolds.

Every integral current in a locally compact metric space X can be approximated by a Lipschitz chain with respect to the normal mass, provided that Lipschitz maps into X can be extended slightly.

Given a sample of an abstract manifold immersed in some Euclidean space, we describe a way to recover the singular homology of the original manifold. It consists in estimating its tangent bundle—seen as subset of another Euclidean space—from a measure theoretical point of view, and in applying measure-based filtrations for persistent homology. The construction we propose is consistent and stable, and does not require the dimension of the manifold. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to study average Hewitt-Stromberg and box dimensions of typical compact metric spaces belonging to the Gromov-Hausdorff space equipped with the Gromov-Hausdorff metric. [ABSTRACT FROM AUTHOR]

Via the method of interior approximation, we prove trace and extension results for BV functions defined on a bounded domain Ω ⊂ R n such that Ω satisfies 0.1 H n - 1 (∂ Ω \ Ω 0) < ∞ , where H n - 1 is the (n - 1) dimensional Hausdorff measure and Ω 0 is the measure-theoretic exterior of Ω . Stronger results are obtained on a subclass of such domains, which are outward minimizing domains. We also obtain new weak regularity results for generalized Cheeger sets as byproducts. [ABSTRACT FROM AUTHOR]

19 S. K. Smirnov, Decomposition of solenoidal vector charges into elementary solenoids, and the structure of normal one-dimensional flows, Algebra i Analiz 5 (1993), no. 4, 206-238. Anal. 48 (2016), no. 1, 1-33. 6 S. Bianchini and N. A. Gusev, Steady nearly incompressible vector fields in two-dimension: Chain rule and renormalization, Arch. Ration. Keywords: Vector-valued measures; divergence-free measures; superposition Principle; 49Q15; 58A25 EN Vector-valued measures divergence-free measures superposition Principle 49Q15 58A25 879 911 33 10/03/22 20221001 NES 221001 1 Introduction In this paper we study the structure of vector-valued Borel owledgements The first author thanks Graziano Crasta and Annalisa Malusa for interesting discussions during the preparation of the paper; he also kindly acknowledges Gian Paolo Leonardi for introducing him to the problem of rigidity of divergence-free vector fields. [Extracted from the article]

8 V. Caselles, On the entropy conditions for some flux limited diffusion equations, J. Differential Equations 250 (2011), no. 8, 3311-3348. 18 (2018), no. 1, 1-28. 30 G. P. Leonardi and G. Saracco, The prescribed mean curvature equation in weakly regular domains, NoDEA Nonlinear Differential Equations Appl. 25 (2018), no. 2, Paper No. 9. 31 G. P. Leonardi and G. Saracco, Rigidity and trace properties of divergence-measure vector fields, Adv. Calc. Keywords: Divergence-measure vector fields; functions of bounded variation; coarea formula; Gauss-Green formula; semicontinuity; 26B30; 49Q15; 49J45 EN Divergence-measure vector fields functions of bounded variation coarea formula Gauss-Green formula semicontinuity 26B30 49Q15 49J45 787 810 24 10/03/22 20221001 NES 221001 1 cations, Ann. [Extracted from the article]

The optimisation of nonsmooth, nonconvex functions without access to gradients is a particularly challenging problem that is frequently encountered, for example in model parameter optimisation problems. Bilevel optimisation of parameters is a standard setting in areas such as variational regularisation problems and supervised machine learning. We present efficient and robust derivative-free methods called randomised Itoh–Abe methods. These are generalisations of the Itoh–Abe discrete gradient method, a well-known scheme from geometric integration, which has previously only been considered in the smooth setting. We demonstrate that the method and its favourable energy dissipation properties are well defined in the nonsmooth setting. Furthermore, we prove that whenever the objective function is locally Lipschitz continuous, the iterates almost surely converge to a connected set of Clarke stationary points. We present an implementation of the methods, and apply it to various test problems. The numerical results indicate that the randomised Itoh–Abe methods can be superior to state-of-the-art derivative-free optimisation methods in solving nonsmooth problems while still remaining competitive in terms of efficiency. [ABSTRACT FROM AUTHOR]

19 S. K. Smirnov, Decomposition of solenoidal vector charges into elementary solenoids, and the structure of normal one-dimensional flows, Algebra i Analiz 5 (1993), no. 4, 206-238. Anal. 48 (2016), no. 1, 1-33. 6 S. Bianchini and N. A. Gusev, Steady nearly incompressible vector fields in two-dimension: Chain rule and renormalization, Arch. Ration. Keywords: Vector-valued measures; divergence-free measures; superposition Principle; 49Q15; 58A25 EN Vector-valued measures divergence-free measures superposition Principle 49Q15 58A25 879 911 33 10/03/22 20221001 NES 221001 1 Introduction In this paper we study the structure of vector-valued Borel owledgements The first author thanks Graziano Crasta and Annalisa Malusa for interesting discussions during the preparation of the paper; he also kindly acknowledges Gian Paolo Leonardi for introducing him to the problem of rigidity of divergence-free vector fields. [Extracted from the article]

This paper proposes a new mathematical and computational tool for inferring the geometry of shapes known only through approximations such as triangulated or digital surfaces. The main idea is to decouple the position of the shape boundary from its normal vector field. To do so, we extend a classical tool of geometric measure theory, the normal cycle, so that it takes as input not only a surface but also a normal vector field. We formalize it as a current in the oriented Grassmann bundle R 3 × S 2 . By choosing adequate differential forms, we define geometric measures like area, mean and Gaussian curvatures. We then show the stability of these measures when both position and normal input data are approximations of the underlying continuous shape. As a byproduct, our tool is able to correctly estimate curvatures over polyhedral approximations of shapes with explicit bounds, even when their natural normal are not correct, as long as an external convergent normal vector field is provided. Finally, the accuracy, convergence and stability under noise perturbation is evaluated experimentally onto digital surfaces. [ABSTRACT FROM AUTHOR]

In this paper, we start a detailed study of a new notion of rectifiability in Carnot groups: we say that a Radon measure is P h -rectifiable, for h ∈ N , if it has positive h-lower density and finite h-upper density almost everywhere, and, at almost every point, it admits a unique tangent measure up to multiples. First, we compare P h -rectifiability with other notions of rectifiability previously known in the literature in the setting of Carnot groups, and we prove that it is strictly weaker than them. Second, we prove several structure properties of P h -rectifiable measures. Namely, we prove that the support of a P h -rectifiable measure is almost everywhere covered by sets satisfying a cone-like property, and in the particular case of P h -rectifiable measures with complemented tangents, we show that they are supported on the union of intrinsically Lipschitz and differentiable graphs. Such a covering property is used to prove the main result of this paper: we show that a P h -rectifiable measure has almost everywhere positive and finite h-density whenever the tangents admit at least one complementary subgroup. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove a Marstrand-type theorem to give the almost sure Hewitt–Stromberg and certain modified intermediate dimensions of projections of sets. We give alternative definitions of dimension profiles in terms of capacities of a set E with respect to certain kernels, which lead to the Hewitt–Stromberg and modified intermediate dimensions of projections fairly easily. [ABSTRACT FROM AUTHOR]

We investigate variants of a three circles type theorem in the context of $\mathcal{Q}-$valued functions. We prove some convexity inequalities related to the $L^{2}$-growth function in the $\mathcal{Q}-$valued settings. Optimality of these inequalities and comparsion to the case of real valued harmonic functions is also discussed. [ABSTRACT FROM AUTHOR]