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Antonella Ritorto

Publications and source records attributed to Antonella Ritorto.

7 recordsLinked to original sources
arXiv2022

Cartesian convexity as the key notion in the variational existence theory for nonlocal supremal functionals

Motivated by the direct method in the calculus of variations in $L^{\infty}$, our main result identifies the notion of convexity characterizing the weakly$^*$ lower semicontinuity of nonlocal supremal functionals: Cartesian level convexity. This new concept coincides with separate level convexity in the one-dimensional setting and is strictly weaker for higher dimensions. We discuss relaxation in the vectorial case, showing that the relaxed functional will not generally maintain the supremal form. Apart from illustrating this fact with examples of multi-well type, we present precise criteria for structure-preservation. When the structure is preserved, a representation formula is given in terms of the Cartesian level convex envelope of the (diagonalized) original supremand. This work does not only complete the picture of the analysis initiated in [Kreisbeck \& Zappale, Calc.~Var.~PDE, 2020], but also establishes a connection with double integrals. We relate the two classes of functionals via an $L^p$-approximation in the sense of $Γ$-convergence for diverging integrability exponents. The proofs exploit recent results on nonlocal inclusions and their asymptotic behavior, and use tools from Young measure theory and convex analysis.

math.AP↗
arXiv2021

Asymptotic analysis of deformation behavior in high-contrast fiber-reinforced materials: Rigidity and anisotropy

We identify the restricted class of attainable effective deformations in a model of reinforced composites with parallel, long, and fully rigid fibers embedded in an elastic body. In mathematical terms, we characterize the weak limits of sequences of Sobolev maps whose gradients on the fibers lie in the set of rotations. These limits are determined by an anisotropic constraint in the sense that they locally preserve length in the fiber direction. Our proof of the necessity emerges as a natural generalization and modification of the recently established asymptotic rigidity analysis for composites with layered reinforcements. However, the construction of approximating sequences is more delicate here due to the higher flexibility and connectedness of the soft material component. We overcome these technical challenges by a careful approximation of the identity that is constant on the rigid components, combined with a lifting in fiber bundles for Sobolev functions. The results are illustrated with several examples of attainable effective deformations. If an additional second-order regularization is introduced into the material model, only rigid body motions can occur macroscopically.

math.AP↗
arXiv2019

A minimization problem involving a fractional Hardy-Sobolev type inequality

In this work, we obtain an existence of nontrivial solutions to a minimization problem involving a fractional Hardy-Sobolev type inequality in the case of inner singularity. Precisely, for $λ>0$ we analyze the attainability of the optimal constant $$ μ_{α, λ}(Ω):=\inf\left\{ [u]^2_{s,Ω}+λ\int_Ω|u|^2 \, dx \colon u\in H^s(Ω), \, \int_Ω \frac{|u(x)|^{2_{s,α}}}{|x|^α} \, dx=1 \right\}, $$ where $0 4s, 0<α<2s$, $2_{s,α}=\frac{2(n-α)}{n-2s}$, and $Ω\subset \mathbb{R}^n$ be a bounded domain such that $0\in Ω$.

math.AP↗
arXiv2019

Nonnegative solutions for the fractional Laplacian involving a nonlinearity with zeros

We study the nonlocal nonlinear problem \begin{equation}\label{ppp} \left\{ \begin{array}[c]{lll} (-Δ)^s u = λf(u) & \mbox{in }Ω, \\ u=0&\mbox{on } \mathbb{R}^N\setminusΩ, \end{array} \right. \tag{$P_λ$} \end{equation} where $Ω$ is a bounded smooth domain in $\mathbb{R}^N$\!,\,$N>2s$,\,$0<s<1$; $f:\mathbb{R}\rightarrow [0,\infty)$ is a nonlinear continuous function such that $f(0)=f(1)=0$ and $f(t)\sim |t|^{p-1}t$ as $t\rightarrow 0^+$, with $2<p+1<2^*_s$; and $λ$ is a positive parameter. We prove the existence of two nontrivial solutions $u_λ$ and $v_λ$ to (\ref{ppp}) such that $0\le u_λ< v_λ\le 1$ for all sufficiently large $λ$. The first solution $u_λ$ is obtained by applying the Mountain Pass Theorem, whereas the second, $v_λ$, via the sub- and super-solution method. We point out that our results hold regardless of the behavior of the nonlinearity $f$ at infinity. In addition, we obtain that these solutions belong to $L^{\infty}(Ω)$.

math.AP↗
arXiv2017

Optimal partition problems for the fractional laplacian

In this work, we prove an existence result for an optimal partition problem of the form $$\min \{F_s(A_1,\dots,A_m)\colon A_i \in \mathcal{A}_s, \, A_i\cap A_j =\emptyset \mbox{ for } i\neq j\},$$ where $F_s$ is a cost functional with suitable assumptions of monotonicity and lowersemicontinuity, $\mathcal{A}_s$ is the class of admissible domains and the condition $A_i\cap A_j =\emptyset$ is understood in the sense of the Gagliardo $s$-capacity, where $0<s<1$. Examples of this type of problem are related to the fractional eigenvalues. In addition, we prove some type of convergence of the $s$-minimizers to the minimizer of the problem with $s=1$, studied in \cite{Bucur-Buttazzo-Henrot}.

math.AP↗
arXiv2016

Shape optimization problems for nonlocal operators

In this work we study a general shape optimization problem where the state equation is given in terms of a nonlocal operator. Examples of the problems considered are monotone combinations of fractional eigenvalues. Moreover, we also analyze the transition from nonlocal to local state equations.

math.AP↗