Search arXiv⌕ Search

arXiv subjects

Matteo Novaga

Publications and source records attributed to Matteo Novaga.

At least 19 recordsLinked to original sources
arXiv2026

The fractional Faber--Krahn inequality in sharp quantitative form

We prove that the first Dirichlet eigenvalue of the fractional Laplacian satisfies the sharp quantitative Faber--Krahn inequality. This was conjectured by Brasco, Cinti and Vita, who proved the inequality with a sub-optimal exponent. The proof adapts to the non-local setting the level-set argument recently introduced by Guerra, Machado and Ramos for the Laplacian. It is carried out on the horizontal level sets of the Caffarelli--Silvestre extension of the torsion function, where the asymmetry appears only as a boundary value at the corner of the extension variables.

math.AP↗
arXiv2026

Second variation of the fractional torsion energy and linear stability of the ball

We compute the second shape derivative of the fractional torsion. It is an explicit quadratic form on the boundary, which involves a Dirichlet-to-Neumann operator for half-harmonic functions blowing up at the boundary. We deduce an explicit formula for volume preserving variations on the ball and show that as in the local case, the shape Hessian is negative and coercive in \(H^{1/2}\) modulo translations.

math.AP↗
arXiv2026

The isoperimetric problem for four-dimensional parallelohedra

The honeycomb and Kelvin problems ask how space can be partitioned into cells of equal volume with the least surface area. We resolve this question in four dimensions for convex bodies that tile space by translations, showing that the regular $24$-cell uniquely minimizes surface area at fixed volume. We develop a framework combining an exact deficit identity, valid in every dimension, with a descent argument. As a consequence, the regular $24$-cell also minimizes surface area among such tiles containing a fixed ball.

math.MG↗
arXiv2026

The rhombic dodecahedral conjecture

We prove that every three-dimensional parallelohedron $P$ satisfies \[ \mathcal H^2(\partial P)\ge 12\sqrt2\,r(P)^2, \] where $r(P)$ is the inradius of $P$, with equality if and only if $P$ is similar to the regular rhombic dodecahedron.

math.MG↗
arXiv2026

The truncated octahedron minimizes surface area among parallelohedra of equal volume

We prove that the regular truncated octahedron uniquely minimizes surface area among all parallelohedra of fixed volume. Equivalently, every three-dimensional parallelohedron $P$ satisfies \[ \frac{\mathcal H^2(\partial P)}{|P|^{2/3}} \ge \frac{3(1+2\sqrt3)}{4^{2/3}}, \] with equality if and only if $P$ is similar to the regular truncated octahedron. Among the non-truncated Fedorov types we prove a stronger sharp bound, attained uniquely by the regular rhombic dodecahedron.

math.MG↗
arXiv2026

Anisotropic isoperimetric double tilings of the plane

We study periodic partitions of the plane into two distinct cells minimizing the anisotropic $\ell_1$-perimeter. For a rectangular lattice $G$, we compute explicitly the $(G,\ell_1)$-isoperimetric profile and classify all minimizers. When one cell has small area, the optimal tiling is generated by a square and a chipped rectangle, while in the remaining regime the tiling is generated by two adjacent rectangles. Further minimizing over all possible planar lattices, we show that the $\ell_1$-isoperimetric profile is attained by the Pythagorean double tiling of two axis-aligned squares sharing a vertex. This configuration is unique unless the two cells are assigned the same area. Finally, we prove that the limiting (non periodic) partitions obtained by sending one volume to infinity are locally $\ell_1$-isoperimetric.

math.AP↗
arXiv2026

On the independence number of de Bruijn graphs

We derive the asymptotic formula $α(k,q)=λ_{k-1}q^k+o(q^k)$, where $α(k,q)$ is the independence number of the de Bruijn graph $B(k,q)$, and $λ_{k-1}$ is a constant arising from a variational problem on the unit $(k-1)$-dimensional cube. When $k=4$, we show the bounds $91/240\le λ_3\le 11/28$. For odd prime $k$, we analyse the binary case $q=2$ via a phase reduction on rotation orbits. For $k=11,13,17$ this yields compact orbit-marker certificates for optimal constructions. Combined with a lifting theorem by Lichiardopol, these certificates give exact formulas for $α(11,q)$, $α(13,q)$, and $α(17,q)$ for all $q\ge2$, extending the known cases $k=3,5,7$.

math.CO↗
arXiv2026

Concentration effects and $Γ$-limit for the elastica functional for open and closed curves

We study the $Γ$-convergence of a class of elastica-type energies defined on immersed planar curves and depending on a small positive parameter $ε$. As $ε\to 0^+$, sequences with equibounded energy develop concentration phenomena in the curvature, leading to the emergence of singularities described by atomic measures. This naturally gives rise to a limiting framework in terms of pointed curves, consisting of a curve together with a measure encoding curvature concentration. We characterize the first-order $Γ$-limit in two settings: for immersed open curves with fixed endpoints and boundary conditions on the tangents, and for immersed closed curves of prescribed length. In both cases, the limiting energy depends only on the number of concentration points and is expressed as a sum of contributions, each given by an integer multiple of $2π$. A key feature of the problem is that the rescaled energies exhibit a structure closely related to one-dimensional Modica--Mortola type functionals.

math.AP↗
arXiv2026

Local minimality of the truncated octahedron for the isoperimetric problem on parallelohedra

We investigate the isoperimetric problem for the Voronoi cells of three-dimensional lattices. Using Selling parameters, we derive an explicit closed formula for the scale-invariant isoperimetric quotient $F$ in terms of six non-negative variables. We then analyse the local behaviour of $F$ at the most relevant lattice configurations: we prove that the body-centered cubic lattice (BCC) is a strict local minimiser of $F$ at fixed volume, whereas the face-centered cubic lattice (FCC) and the simple cubic lattice (SC) are not local minimisers. Then, we consider a family of lattices which interpolates between BCC and FCC, showing that BCC is the global minimiser of $F$ restricted to this family.

math.MG↗
arXiv2026

Optimal sets for a geometric oscillation energy

We investigate the nonlocal energy corresponding to the $p$-oscillation of the unit normal vector for hypersurfaces, or the unit tangent vector for curves. The energy satisfies geometric inequalities with optimal constants $c(n,p)$ and $C(n,p)$ which are determined by a variational problem over the probability measures on the sphere. The extremal measures for such problem depend critically on the value of $p$. We prove existence of optimal sets for this energy under perimeter and volume constraint, and characterize their shape.

math.AP↗
arXiv2026

Non-local planelike minimizers and $Γ$-convergence of periodic energies to a local anisotropic perimeter

We investigate a homogenization problem related to a non-local interface energy with a periodic forcing term. We show the existence of planelike minimizers for such energy. Moreover, we prove that, under suitable assumptions on the non-local kernel and the external field, the sequence of rescaled energies $Γ$-converges to a suitable local anisotropic perimeter, where the anisotropy is defined as the limit of the normalized energy of a planelike minimizer in larger and larger cubes (i.e., what is called in jargon "stable norm"). To obtain this, we also establish several auxiliary results, including: the minimality of the level sets of the minimizers, explicit bounds on the oscillations of the minimizers, density estimates for almost minimizers, and non-local perimeter estimates in the large.

math.AP↗
arXiv2026

A fourth-order regularization of the curvature flow of immersed plane curves with Dirichlet boundary conditions

We consider a fourth-order regularization of the curvature flow for an immersed plane curve with fixed boundary, using an elastica-type functional depending on a small positive parameter $\varepsilon$. We show that the approximating flow smoothly converges, as $\varepsilon \to 0^+$, to the curvature flow of the curve with Dirichlet boundary conditions for all times before the first singularity of the limit flow.

math.AP↗
arXiv2025

Quantitative estimates for bending energies and applications to non-local variational problems

We discuss a variational model, given by a weighted sum of perimeter, bending and Riesz interaction energies, that could be considered as a toy model for charged elastic drops. The different contributions have competing preferences for strongly localized and maximally dispersed structures. We investigate the energy landscape in dependence of the size of the 'charge', i.e. the weight of the Riesz interaction energy. In the two-dimensional case we first prove that for simply connected sets of small elastic energy, the elastic deficit controls the isoperimetric deficit. Building on this result, we show that for small charge the only minimizers of the full variational model are either balls or centered annuli. We complement these statements by a non-existence result for large charge. In three dimensions, we prove area and diameter bounds for configurations with small Willmore energy and show that balls are the unique minimizers of our variational model for sufficiently small charge.

math.AP↗
arXiv2025

Euler's elastica functional as a large mass limit of a two-dimensional non-local isoperimetric problem

We consider a large mass limit of the non-local isoperimetric problem with a repulsive Yukawa potential in two space dimensions. In this limit, the non-local term concentrates on the boundary, resulting in the existence of a critical regime in which the perimeter and the non-local terms cancel each other out to leading order. We show that under appropriate scaling assumptions the next-order $Γ$-limit of the energy with respect to the $L^1$ convergence of the rescaled sets is given by a weighted sum of the perimeter and Euler's elastica functional, where the latter is understood via the lower-semicontinuous relaxation and is evaluated on the system of boundary curves. As a consequence, we prove that in the considered regime the energy minimizers always exist and converge to either disks or annuli, depending on the relative strength of the elastica term.

math.AP↗
arXiv2025

Periodic double tilings of the plane

We study tilings of the plane composed of two repeating tiles of different assigned areas relative to an arbitrary periodic lattice. We classify isoperimetric configurations (i.e., configurations with minimal length of the interfaces) both in the case of a fixed lattice or for an arbitrary periodic lattice. We find three different configurations depending on the ratio between the assigned areas of the two tiles and compute the isoperimetric profile. The three different configurations are composed of tiles with a different number of circular edges, moreover, different configurations exhibit a different optimal lattice. Finally, we raise some open problems related to our investigation.

math.MG↗