arXiv subjects

Giovanni Di Fratta

Publications and source records attributed to Giovanni Di Fratta.

At least 19 recordsLinked to original sources

arXiv2026

A Fixed-Point Construction of the Elementary Transcendental Functions

We present a unified fixed-point construction of the elementary transcendental functions, encompassing the real exponential, the complex exponential (sine and cosine), and the natural logarithm. Each function is characterized as the unique solution of a duplication identity established through the Banach contraction principle. These foundational identities are $e(2x)=e^2(x)$ for the exponentials, and $\log(x^{2})=2\log x$ for the logarithm. Because a direct iteration of these identities is numerically unstable, owing to local expansiveness at the target, the central idea transfers the analysis to a residual function, on which the operator becomes a strict contraction with an explicit convergence rate. Beyond its theoretical economy, which dispenses with differential equations and power series, this characterization translates into efficient algorithms for the machine evaluation of elementary functions: the underlying framework yields floating-point kernels whose accuracy and iteration depth are governed by the theoretical contraction rate. We also present a computational study showing that, in a throughput-bound vectorized regime, these kernels are competitive with standard production libraries, and in favorable configurations exceed them, with the sine--cosine kernel faster at every tested iteration depth. These implementations operate without lookup tables or memory traffic, an architectural advantage for modern high-performance and energy-efficient computing.

math.CA↗

arXiv2026

Variational principles for the interaction of liquid crystals and electric fields in the Oseen--Frank model

We develop a rigorous variational framework for uniaxial nematic liquid crystals interacting with an external electric field in the one-constant Oseen--Frank approximation. Equilibrium configurations are governed by a nonlocal, nonlinear energy with a challenging min-max saddle-point structure. Our first main result reformulates this problem as a pure double-minimization problem. Using convex duality and the Hodge decomposition, we replace the scalar electrostatic potential with a vector potential, yielding a closed-form dual functional with a unique minimizer. This direct energy-minimization principle is advantageous for both theoretical analysis and numerical simulation. Our second result rigorously quantifies the decoupling of the electrostatic back-reaction in the limit of small dielectric anisotropy. We establish a uniform quadratic energy bound between the exact nonlocal energy and its standard local approximation, formally justifying the widespread physics heuristic of neglecting induced depolarization fields. Finally, under a strict coercivity condition, we combine $Γ$-convergence, uniform Sobolev regularity, and a perturbative coercivity transfer to prove that physical minimizers converge to limiting harmonic maps at an optimal, quantitative linear rate.

math.AP↗

arXiv2026

A Classical Elliptic Regularity Approach to Almost Harmonic Maps and Related Systems

We develop an abstract regularity framework for a class of two-dimensional nonlinear elliptic systems, including almost harmonic maps. The approach combines a Campanato-type iteration scheme with a Caccioppoli-type estimate and identifies general assumptions under which local H{ö}lder continuity follows. More precisely, we prove that any class of admissible pairs $(u,f)$ that is stable under rescaling and satisfies an oscillation-decay property consists of locally H{ö}lder continuous maps. The resulting H{ö}lder exponent is explicit and matches the classical Morrey--Campanato threshold determined by the Lebesgue integrability of the source term $f$. The framework is purely analytic and avoids the use of $\mathcal{H}^1$--$\mathrm{BMO}$ duality, Wente's inequality, moving frames, and conformal uniformization. We illustrate the flexibility of the framework through several classes of elliptic systems. As a first example, we recover local H{ö}lder continuity for almost harmonic maps \[ -Δu=|\nabla u|^2u+f \] into $\mathbb{S}^n$ with $L^q$-integrable tension fields by means of a direct argument independent of the classical harmonic map regularity theory. We next consider systems of the form \[ -Δu=Ω\cdot\nabla u+f, \] showing that the analytic condition $\operatorname{div}Ω\in L^q$ for some $q>1$ is sufficient to ensure regularity (for classical harmonic maps, $\operatorname{div}Ω=0$). We further apply the framework to...

math.AP↗

arXiv2025

Necessary and Sufficient Conditions for the Maz'ya-Shaposhnikova Formula in (Fractional) Sobolev Spaces

We investigate the asymptotic behavior, as $\varepsilon \to 0$, of nonlocal functionals $$ \mathcal{F}_{\varepsilon}(u) = \iint_{\mathbb{R}^N\times\mathbb{R}^N} ρ_{\varepsilon}(y-x)\,|u(x)-u(y)|^p\,dx\,dy,\qquad u\in L^p(\mathbb{R}^N),\quad 1\leqslant p<\infty, $$ associated with a general family of nonnegative measurable kernels $\{ρ_{\varepsilon}\}_{\varepsilon>0}$. Our primary aim is to single out the weakest moment-type assumptions on the family $\{ρ_{\varepsilon}\}_{\varepsilon>0}$ that are necessary and sufficient for the pointwise convergence $$ \lim_{\varepsilon\to 0}\mathcal{F}_{\varepsilon}(u)=2\|u\|_{L^p}^p $$ to hold for every $u$ in a prescribed subspace of $L^p(\mathbb{R}^N)$. In the canonical smooth regime of compactly supported functions ($u\in C_c^{\infty}(\mathbb{R}^N)$) we show that convergence occurs when two optimal conditions are satisfied: (i) a mass-escape condition, and (ii) a short-range attenuation effect, expressed by the vanishing as $\varepsilon\to 0$ of the kernels' $p$-moments in any fixed neighborhood of the origin. This general framework recovers the classical Maz'ya--Shaposhnikova theorem for fractional-type kernels and extends the convergence result to a much broader class of interaction profiles, which may be non-symmetric and non-homogeneous. Using a density argument that preserves the moment assumptions, we prove that the same necessary and sufficient conditions remain valid in the integer-order Sobolev setting ($u\in W^{1,p}(\mathbb{R}^N)$). Finally, by adapting the method to fractional Sobolev spaces $W^{s,p}(\mathbb{R}^N)$ with $s\in(0,1)$, we recover the Maz'ya-Shaposhnikova formula and extend it under analogous abstract conditions on the family $\{ρ_{\varepsilon}\}_{\varepsilon>0}$.

math.AP↗

arXiv2025

Energy minimization for skyrmions on planar thin films

We consider an energy functional that arises in micromagnetic and liquid crystal theory on thin films. In particular, our energy comprises a non-convex term that models anti-symmetric exchange as well as an anisotropy term. We devise an algorithm for energy minimization in the continuous case and show weak convergence of a subsequence towards a solution of the corresponding Euler--Lagrange equation. Furthermore, an algorithm for numerical energy minimization is presented. We show empirically that this numerical algorithm converges to the correct solutions for a benchmark problem without the need for user-supplied parameters, and present a rigorous convergence analysis for important special cases.

math.NA↗

arXiv2025

Nonlocal Micromagnetics: Compactness Criteria, Existence of Minimizers, and Brown's Fundamental Theorem

This paper investigates the existence and qualitative properties of minimizers for a class of nonlocal micromagnetic energy functionals defined on bounded domains. The considered energy functional consists of a symmetric exchange interaction, which penalizes spatial variations in magnetization, and a magnetostatic self-energy term that accounts for long-range dipolar interactions. Motivated by the extension of Brown's fundamental theorem on fine ferromagnetic particles to nonlocal settings, we develop a rigorous variational framework in $L^2(Ω;\mathbb{S}^2)$ under mild assumptions on the interaction kernel $j$, including symmetry, Lévy-type integrability, and prescribed singular behavior. For spherical domains, we generalize Browns fundamental results by identifying critical radii $R^*$ and $R^{**}$ that delineate distinct energetic regimes: for $ R \leq R^*$, the uniform magnetization state is energetically preferable (small-body regime), whereas for $R \geq R^{**}$, non-uniform magnetization configurations become dominant (large-body regime). These transitions are analyzed through Poincaré-type inequalities and explicit energy comparisons between uniform and vortex-like magnetization states. Our results directly connect classical micromagnetic theory and contemporary nonlocal models, providing new insights into domain structure formation in nanoscale magnetism. Furthermore, the mathematical framework developed in this work contributes to advancing theoretical foundations for applications in spintronics and data storage technologies.

math.AP↗

arXiv2024

Korn and Poincaré-Korn inequalities: A different perspective

We present a concise point of view on the first and the second Korn's inequality for general exponent $p$ and for a class of domains that includes Lipschitz domains. Our argument is conceptually very simple and, for $p = 2$, uses only the classical Riesz representation theorem in Hilbert spaces. Moreover, the argument for the general exponent $1<p<\infty$ remains the same, the only change being invoking now the $q$-Riesz representation theorem (with $q$ the harmonic conjugate of $p$). We also complement the analysis with elementary derivations of Poincaré-Korn inequalities in bounded and unbounded domains, which are essential tools in showing the coercivity of variational problems of elasticity but also propedeutic to the proof of the first Korn inequality.

math.AP↗

arXiv2024

Reduced theory of symmetric and antisymmetric exchange interactions in nanowires

We investigate the behavior of minimizers of perturbed Dirichlet energies supported on a wire generated by a regular simple curve $γ$ and defined in the space of $\mathbb{S}^2$-valued functions. The perturbation $K$ is represented by a matrix-valued function defined on $\mathbb{S}^2$ with values in $\mathbb{R}^{3 \times 3}$. Under natural regularity conditions on $K$, we show that the family of perturbed Dirichlet energies converges, in the sense of $Γ$-convergence, to a simplified energy functional on $γ$. The reduced energy unveils how part of the antisymmetric exchange interactions contribute to an anisotropic term whose specific shape depends on the curvature of $γ$. We also discuss the significant implications of our results for studies of ferromagnetic nanowires when Dzyaloshinskii-Moriya interaction (DMI) is present.

math.AP↗

arXiv2024

A modular Poincaré-Wirtinger type inequality on Lipschitz domains for Sobolev spaces with variable exponents

In the context of Sobolev spaces with variable exponents, Poincaré--Wirtinger inequalities are possible as soon as Luxemburg norms are considered. On the other hand, modular versions of the inequalities in the expected form \begin{equation*} \int_Ω\left|f(x)-\langle f\rangle_Ω\right|^{p(x)} \ {\mathrm{d} x} \leqslant C \int_Ω|\nabla f(x)|^{p(x)}{\mathrm{d} x}, \end{equation*} are known to be \emph{false}. As a result, all available modular versions of the Poincaré- Wirtinger inequality in the variable-exponent setting always contain extra terms that do not disappear in the constant exponent case, preventing such inequalities from reducing to the classical ones in the constant exponent setting. Our contribution is threefold. First, we establish that a modular Poincaré--Wirtinger inequality particularizing to the classical one in the constant exponent case is indeed conceivable. We show that if $Ω\subset \mathbb{R}^n$ is a bounded Lipschitz domain, and if $p\in L^\infty(Ω)$, $p \geq 1$, then for every $f\in C^\infty(\barΩ)$ the following generalized Poincaré--Wirtinger inequality holds \begin{equation*} \int_Ω\left|f(x)-\langle f\rangle_Ω\right|^{p(x)} \ {\mathrm{d} x} \leq C \int_Ω\int_Ω\frac{|\nabla f(z)|^{p(x)}}{|z-x|^{n-1}}\ {\mathrm{d} z}{\mathrm{d} x}, \end{equation*} where $\langle f\rangle_Ω$ denotes the mean of $f$ over $Ω$, and $C>0$ is a positive constant depending only on $Ω$ and $\|p\|_{L^\infty(Ω)}$. Second, our argument is concise and constructive and does not rely on compactness results. Third, we additionally provide geometric information on the best Poincaré--Wirtinger constant on Lipschitz domains.

math.AP↗

arXiv2024

A Bourgain-Brezis-Mironescu formula accounting for nonlocal antisymmetric exchange interactions

The present study concerns the nonlocal-to-local convergence of a family of exchange energy functionals in the limit of very short-range interactions. The analysis accounts for both symmetric and antisymmetric exchange. Our result is twofold. First, we extend the Bourgain-Brezis-Mironescu formula to encompass the scenario where antisymmetric contributions are encoded into the energy. Second, we prove that, under physically relevant assumptions on the families of exchange kernels, the family of nonlocal functionals Gamma-converges to their local counterparts. As a byproduct of our analysis, we obtain a rigorous justification of Dzyaloshinskii-Moriya interactions in chiral magnets under the form commonly adopted in the variational theory of micromagnetism when modeling antisymmetric exchange interactions.

math.AP↗

arXiv2023

Sufficient conditions for the existence of minimizing harmonic maps with axial symmetry in the small-average regime

The paper concerns the analysis of global minimizers of a Dirichlet-type energy functional defined on the space of vector fields $H^1(S,T)$, where $S$ and $T$ are surfaces of revolution. The energy functional we consider is closely related to a reduced model in the variational theory of micromagnetism for the analysis of observable magnetization states in curved thin films. We show that axially symmetric minimizers always exist, and if the target surface $T$ is never flat, then any coexisting minimizer must have line symmetry. Thus, the minimization problem reduces to the computation of an optimal one-dimensional profile. We also provide a necessary and sufficient condition for energy minimizers to be axially symmetric.

math.AP↗

arXiv2023

Sharp conditions for the validity of the Bourgain-Brezis-Mironescu formula

Following the seminal paper by Bourgain, Brezis and Mironescu, we focus on the asymptotic behavior of some nonlocal functionals that, for each $u\in L^2(\mathbb{R}^N)$, are defined as the double integrals of weighted, squared difference quotients of $u$. Given a family of weights $\{ρ_ε\}$, $ε\in (0,1)$, we devise sufficient and necessary conditions on $\{ρ_ε\}$ for the associated nonlocal functionals to converge as $ε\to 0$ to a variant of the Dirichlet integral. Finally, some comparison between our result and the existing literature is provided.

math.AP↗

arXiv2022

Curved thin-film limits of chiral Dirichlet energies

We investigate the curved thin-film limit of a family of perturbed Dirichlet energies in the space of $H^1$ Sobolev maps defined in a tubular neighborhood of an $(n - 1)$-dimensional submanifold $N$ of $\mathbb{R}^n$ and with values in an $(m - 1)$-dimensional submanifold $M$ of $\mathbb{R}^m$. The perturbation $\mathsf{K}$ that we consider is represented by a matrix-valued function defined on $M$ and with values in $\mathbb{R}^{m \times n}$. Under natural regularity hypotheses on $N$, $M$, and $\mathsf{K}$, we show that the family of these energies converges, in the sense of $Γ$-convergence, to an energy functional on $N$ of an unexpected form, which is of particular interest in the theory of magnetic skyrmions. As a byproduct of our results, we get that in the curved thin-film limit, antisymmetric exchange interactions also manifest under an anisotropic term whose specific shape depends both on the curvature of the thin film and the curvature of the target manifold. Various types of antisymmetric exchange interactions in the variational theory of micromagnetism are a source of inspiration and motivation for our work.

math.AP↗

arXiv2022

On symmetry of energy minimizing harmonic-type maps on cylindrical surfaces

The paper concerns the analysis of global minimizers of a Dirichlet-type energy functional in the class of $\mathbb{S}^2$-valued maps defined in cylindrical surfaces. The model naturally arises as a curved thin-film limit in the theories of nematic liquid crystals and micromagnetics. We show that minimal configurations are $z$-invariant and that energy minimizers in the class of weakly axially symmetric competitors are, in fact, axially symmetric. Our main result is a family of sharp Poincaré-type inequality on the circular cylinder, which allows establishing a nearly complete picture of the energy landscape. The presence of symmetry-breaking phenomena is highlighted and discussed. Finally, we provide a complete characterization of in-plane minimizers, which typically appear in numerical simulations for reasons we explain.

math.AP↗

arXiv2022

The mass-lumped midpoint scheme for computational micromagnetics: Newton linearization and application to magnetic skyrmion dynamics

We discuss a mass-lumped midpoint scheme for the numerical approximation of the Landau-Lifshitz-Gilbert equation, which models the dynamics of the magnetization in ferromagnetic materials. In addition to the classical micromagnetic field contributions, our setting covers the non-standard Dzyaloshinskii-Moriya interaction, which is the essential ingredient for the enucleation and stabilization of magnetic skyrmions. Our analysis also includes the inexact solution of the arising nonlinear systems, for which we discuss both a constraint preserving fixed-point solver from the literature and a novel approach based on the Newton method. We numerically compare the two linearization techniques and show that the Newton solver leads to a considerably lower number of nonlinear iterations. Moreover, in a numerical study on magnetic skyrmions, we demonstrate that, for magnetization dynamics that are very sensitive to energy perturbations, the midpoint scheme, due to its conservation properties, is superior to the dissipative tangent plane schemes from the literature.

math.NA↗

arXiv2021

Symmetry properties of minimizers of a perturbed Dirichlet energy with a boundary penalization

We consider $\mathbb{S}^2$-valued maps on a domain $Ω\subset\mathbb{R}^N$ minimizing a perturbation of the Dirichlet energy with vertical penalization in $Ω$ and horizontal penalization on $\partialΩ$. We first show the global minimality of universal constant configurations in a specific range of the physical parameters using a Poincaré-type inequality. Then, we prove that any energy minimizer takes its values into a fixed meridian of the sphere $\mathbb{S}^2$, and deduce uniqueness of minimizers up to the action of the appropriate symmetry group. We also prove a comparison principle for minimizers with different penalizations. Finally, we apply these results to a problem on a ball and show radial symmetry and monotonicity of minimizers. In dimension $N=2$ our results can be applied to the Oseen--Frank energy for nematic liquid crystals and micromagnetic energy in a thin-film regime.

math.AP↗

arXiv2021

A unified divergent approach to Hardy-Poincaré inequalities in classical and variable Sobolev spaces

We present a unified strategy to derive Hardy-Poincaré inequalities on bounded and unbounded domains. The approach allows proving a general Hardy-Poincaré inequality from which the classical Poincaré and Hardy inequalities immediately follow. The idea also applies to the more general context of variable exponent Sobolev spaces. The argument, concise and constructive, does not require a priori knowledge of compactness results and retrieves geometric information on the best constants.

math.AP↗

arXiv2020

Micromagnetics of thin films in the presence of Dzyaloshinskii-Moriya interaction

In this paper, we study the thin-film limit of the micromagnetic energy functional in the presence of bulk Dzyaloshinskii-Moriya interaction (DMI). Our analysis includes both a stationary $Γ$-convergence result for the micromagnetic energy, as well as the identification of the asymptotic behavior of the associated Landau-Lifshitz-Gilbert equation. In particular, we prove that, in the limiting model, part of the DMI term behaves like the projection of the magnetic moment onto the normal to the film, contributing this way to an increase in the shape anisotropy arising from the magnetostatic self-energy. Finally, we discuss a convergent finite element approach for the approximation of the time-dependent case and use it to numerically compare the original three-dimensional model with the two-dimensional thin-film limit.

math.AP↗