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Emanuele Spadaro

Publications and source records attributed to Emanuele Spadaro.

At least 19 recordsLinked to original sources
arXiv2026

Generic configurations in 2D strongly competing systems

We study a problem modelling segregation of an arbitrary number of competing species in planar domains. The solutions give rise to a well known free boundary problem with the domain partitioning itself into subdomains occupied by different species. In principle, several of them can coexist in a neighborhood of any point. However, we show that {\it generically} the domain partitions into subdomains with only triple junctions, meaning that at most three populations meet at the free boundary. Our main tools are the use of the formalism of harmonic maps into singular spaces and the introduction of a complex structure via the Hopf differential.

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arXiv2024

Clearing-out of dipoles for minimisers of 2-dimensional discrete energies with topological singularities

A key question in the analysis of discrete models for material defects, such as vortices in spin systems and superconductors or isolated dislocations in metals, is whether information on boundary energy for a domain can be sufficient for controlling the number of defects in the interior. We present a general combinatorial dipole-removal argument for a large class of discrete models including XY systems and screw dislocation models, allowing to prove sharp conditions under which controlled flux and boundary energy guarantee to have minimizers with zero or one charges in the interior. The argument uses the max-flow min-cut theorem in combination with an ad-hoc duality for planar graphs, and is robust with respect to changes of the function defining the interaction energies.

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arXiv2024

On the free boundary for thin obstacle problems with Sobolev variable coefficients

We establish a quasi-monotonicity formula {for an intrinsic frequency function related to solutions to} thin obstacle problems with zero obstacle driven by quadratic energies with Sobolev $W^{1,p}$ coefficients, with $p$ bigger than the space dimension. From this we deduce several regularity and structural properties of the corresponding free boundaries at those distinguished points with finite order of contact with the obstacle. In particular, we prove the rectifiability {and the local finiteness of the Minkowski content} of the whole free boundary in the case of Lipschitz coefficients.

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arXiv2024

On the nonlinear thin obstacle problem

The thin obstacle problem or $n$-dimensional Signorini problem is a classical variational problem arising in several applications, starting with its first introduction in elasticity theory. The vast literature concerns mostly quadratic energies, whereas only partial results have been proved in the nonlinear case. In this paper we consider the thin boundary obstacle problem for a general class of nonlineraities and we prove the optimal $C^{1, \frac{1}{2}}$-regularity of the solutions in any space dimension.

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arXiv2023

Parabolic $α$-Riesz flows and limit cases $α\to 0^+$, $α\to d^-$

In this paper we introduce the notion of parabolic $α$-Riesz flow, for $α\in(0,d)$, extending the notion of $s$-fractional heat flows to negative values of the parameter $s=-\fracα{2}$. Then, we determine the limit behaviour of these gradient flows as $α\to 0^+$ and $α\to d^-$. To this end we provide a preliminary $Γ$-convergence expansion for the Riesz interaction energy functionals. Then we apply abstract stability results for uniformly $λ$-convex functionals which guarantee that $Γ$-convergence commutes with the gradient flow structure.

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arXiv2023

On the Read-Shockley energy for grain boundaries in poly-crystals

In the 50's Read and Shockley proposed a formula for the energy of small angle grain boundaries in polycrystals based on linearised elasticity and an ansazt on the distribution of incompatibilities of the lattice at the interface. In this paper we derive a sharp interface limiting functional starting from a nonlinear semidiscrete model for dislocations proposed by Lauteri--Luckhaus. Building upon their analysis we obtain, via $Γ$-convergence, an interfacial energy depending on the rotations of the grains and the relative orientation of the interface which agrees for small angle grain boundaries with the Read and Shockley logarithmic scaling.

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arXiv2022

Two slope functions minimizing fractional seminorms and applications to misfit dislocations

We consider periodic piecewise affine functions, defined on the real line, with two given slopes and prescribed length scale of the regions where the slope is negative. We prove that, in such a class, the minimizers of $s$-fractional Gagliardo seminorm densities, with $0<s<1$, are in fact periodic with the minimal possible period determined by the prescribed slopes and length scale. Then, we determine the asymptotic behavior of the energy density as the ratio between the length of the two intervals where the slope is constant vanishes. Our results, for $s=\frac 1 2$, have relevant applications to the van der Merwe theory of misfit dislocations at semi-coherent straight interfaces. We consider two elastic materials having different elastic coefficients and casting parallel lattices having different spacing. As a byproduct of our analysis, we prove the periodicity of optimal dislocation configurations and we provide the sharp asymptotic energy density in the semi-coherent limit as the ratio between the two lattice spacings tends to one.

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arXiv2021

Regularity of solutions to nonlinear thin and boundary obstacle problems

Variational inequalities with thin obstacles and Signorini-type boundary conditions are classical problems in the calculus of variations, arising in numerous applications. In the linear case many refined results are known, while in the nonlinear setting our understanding is still at a preliminary stage. In this paper we prove $C^1$ regularity for the solutions to a general class of quasi-linear variational inequalities with thin obstacles and $C^{1, α}$ regularity for variational inequalities under Signorini-type conditions on the boundary of a domain.

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arXiv2020

How a minimal surface leaves a thin obstacle

We prove optimal regularity and a detailed analysis of the free boundary of the solutions to the thin obstacle problem for nonparametric minimal surfaces with flat obstacles.

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arXiv2019

The local structure of the free boundary in the fractional obstacle problem

Building upon the recent results in \cite{FoSp17} we provide a thorough description of the free boundary for the fractional obstacle problem in $\mathbb{R}^{n+1}$ with obstacle function $φ$ (suitably smooth and decaying fast at infinity) up to sets of null $\mathcal{H}^{n-1}$ measure. In particular, if $φ$ is analytic, the problem reduces to the zero obstacle case dealt with in \cite{FoSp17} and therefore we retrieve the same results: (i) local finiteness of the $(n-1)$-dimensional Minkowski content of the free boundary (and thus of its Hausdorff measure), (ii) $\mathcal{H}^{n-1}$-rectifiability of the free boundary, (iii) classification of the frequencies and of the blow-ups up to a set of Hausdorff dimension at most $(n-2)$ in the free boundary. Instead, if $φ\in C^{k+1}(\mathbb{R}^n)$, $k\geq 2$, similar results hold only for a distinguished subset of points in the free boundary where the order of contact of the solution and the obstacle is less than $k+1$.

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arXiv2017

Regularity theory for $2$-dimensional almost minimal currents II: branched center manifold

We construct a branched center manifold in a neighborhood of a singular point of a $2$-dimensional integral current which is almost minimizing in a suitable sense. Our construction is the first half of an argument which shows the discreteness of the singular set for the following three classes of $2$-dimensional currents: area minimizing in Riemannian manifolds, semicalibrated and spherical cross sections of $3$-dimensional area minimizing cones.

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arXiv2017

On the measure and the structure of the free boundary of the lower dimensional obstacle problem

We provide a thorough description of the free boundary for the lower dimensional obstacle problem in $\mathbb{R}^{n+1}$ up to sets of null $\mathcal{H}^{n-1}$ measure. In particular, we prove (i) local finiteness of the $(n-1)$-dimensional Hausdorff measure of the free boundary, (ii) $\mathcal{H}^{n-1}$-rectifiability of the free boundary, (iii) classification of the frequencies up to a set of dimension at most (n-2) and classification of the blow-ups at $\mathcal{H}^{n-1}$ almost every free boundary point.

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arXiv2016

On a isoperimetric-isodiametric inequality

The Euclidean mixed isoperimetric-isodiametric inequality states that the round ball maximizes the volume under constraint on the product between boundary area and radius. The goal of the paper is to investigate such mixed isoperimetric-isodiametric inequalities in Riemannian manifolds. We first prove that the same inequality, with the sharp Euclidean constants, holds on Cartan-Hadamard spaces as well as on minimal submanifolds of $\mathbb{R}^n$. The equality cases are also studied and completely characterized; in particular, the latter gives a new link with free boundary minimal submanifolds in a Euclidean ball. We also consider the case of manifolds with non-negative Ricci curvature and prove a new comparison result stating that metric balls in the manifold have product of boundary area and radius bounded by the Euclidean counterpart and equality holds if and only if the ball is actually Euclidean. We then pass to consider the problem of the existence and the regularity of optimal shapes (i.e. regions minimizing the product of boundary area and radius under the constraint of having fixed enclosed volume), called here isoperimetric-isodiametric regions. We give examples of spaces where there exists no isoperimetric-isodiametric region and we prove that on the other hand on compact spaces and on $C^0$-locally-asymptotic Euclidean manifolds with non-negative Ricci curvature there exists an isoperimetric-isodiametric region for every positive volume (this last class of spaces includes a large family of metrics playing a key role in general relativity and Ricci flow: the so called Hawking gravitational instantons and the Bryant-type Ricci solitons). We finally prove that the boundary of the isoperimetric-isodiametric regions satisfy the optimal $C^{1,1}$ regularity outside of a closed subset of Hausdorff co-dimension $8$.

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