arXiv subjects

Luca Briani

Publications and source records attributed to Luca Briani.

8 recordsLinked to original sources

arXiv2026

Essential spectrum for the $p-$Laplacian

We introduce a variational notion of essential spectrum for the Dirichlet $p-$Laplacian. We then extend the classical Persson Theorem to this nonlinear setting. This result provides a geometric characterization of the bottom of the essential spectrum, in terms of the sharp $L^p$ Poincar\'e constant ``at infinity''. We also show that in the case $p=2$ our construction of the essential spectrum is perfectly consistent with the classical theory. Finally, as an example, we compute the full spectrum of the Dirichlet $p-$Laplacian on a rectilinear strip: it is purely essential, with no embedded eigenvalues. The arguments of the proofs are elementary and new already for the linear case $p=2$.

math.AP↗

arXiv2026

Low eigenvalues of the $p-$Laplacian in general open sets

We consider the minmax Ljusternik-Schnirelmann levels of the constrained $p-$Dirichlet integral, on a general open set of the Euclidean space. We show that, whenever one of these levels lies below the threshold given by the $L^p$ Poincar\'e constant ``at infinity'', it actually defines an eigenvalue of the Dirichlet $p-$Laplacian. We also prove an exponential decay at infinity for the relevant eigenfunctions: this can be seen as a \v{S}nol-Simon--type estimate for the nonlinear case. Finally, we exhibit some peculiar examples of unbounded open sets to which our main result applies.

math.AP↗

arXiv2025

Extremals for sharp Poincar\'e-Sobolev inequalities: periodically perforated sets and beyond

We consider periodically perforated unbounded open sets and prove existence of extremals for the relevant sharp Poincar\'e-Sobolev embedding constant. The existence result holds no matter the shape or the regularity of the hole: it is sufficient that the latter is a compact set with positive capacity. We also show how to apply the main result in order to get a similar existence statement, for sets which are periodic in some directions and bounded in all the others.

math.AP↗

arXiv2025

Extremals for Poincar\'e-Sobolev sharp constants in Steiner symmetric sets

We prove existence of minimizers for the sharp Poincar\'e-Sobolev constant in general Steiner symmetric sets, in the subcritical and superhomogeneous regime. The sets considered are not necessarily bounded, thus the relevant embeddings may suffer from a lack of compactness. We prove existence by means of an elementary compactness method. We also prove an exponential decay at infinity for minimizers, showing that in the case of Steiner symmetric sets the relevant estimates only depend on the underlying geometry. Finally, we illustrate the optimality of the existence result, by means of some examples.

math.AP↗

arXiv2025

Energy concentration in a two-dimensional magnetic skyrmion model: variational analysis of lattice and continuum theories

We investigate the formation of singularities in a baby Skyrme type energy model, which describes magnetic solitons in two-dimensional ferromagnetic systems. In presence of a diverging anisotropy term, which enforces a preferred background state of the magnetization, we establish a weak compactness of its topological charge density, which converges to an atomic measure with quantized weights. We characterize the $\Gamma$-limit of the energies as the total variation of this measure. In the case of lattice type energies, we first need to carefully define a notion of discrete topological charge for $\mathbb{S}^2$-valued maps. We then prove a corresponding compactness and $\Gamma$-convergence result, thereby bridging the discrete and continuum theories.

math.AP↗

arXiv2023

Mean-to-max ratio of the torsion function and honeycomb structures

In this paper we study extremal behaviors of the mean to max ratio of the $p$-torsion function with respect to the geometry of the domain. For $p$ larger than the dimension of the space $N$, we prove that the upper bound is uniformly below $1$, contrary to the case $p \in (1,N]$. For $p=+\infty$, in two dimensions, we prove that the upper bound is asymptotically attained by a disc from which is removed a network of points consisting on the vertices of a tiling of the plane with regular hexagons of vanishing size.

math.AP↗

arXiv2022

On a reverse Kohler-Jobin inequality

We consider the shape optimization problems for the quantities $λ(Ω)T^q(Ω)$, where $Ω$ varies among open sets of $\mathbb{R}^d$ with a prescribed Lebesgue measure. While the characterization of the infimum is completely clear, the same does not happen for the maximization in the case $q>1$. We prove that for $q$ large enough a maximizing domain exists among quasi-open sets and that the ball is optimal among {\it nearly spherical domains}.

math.OC↗

arXiv2022

On a class of Cheeger inequalities

We study a general version of the Cheeger inequality by considering the shape functional $\mathcal{F}_{p,q}(Ω)=λ_p^{1/p}(Ω)/λ_q(Ω)^{1/q}$. The infimum and the supremum of $\mathcal{F}_{p,q}$ are studied in the class of all domains $Ω$ of $\mathbb{R}^d$ and in the subclass of convex domains. In the latter case the issue concerning the existence of an optimal domain for $\mathcal{F}_{p,q}$ is discussed.

math.OC↗