arXiv subjects

Marco Picerni

Publications and source records attributed to Marco Picerni.

7 recordsLinked to original sources

arXiv2026

Ideal points, directed completion and the case of the maximally extended Schwarzschild spacetime

We study the directed completion of Lorentzian pre-length spaces, and we show that, under some natural assumptions which cover the case of smooth globally hyperbolic spacetimes, it coincides with the future causal completion of Geroch--Kronheimer--Penrose. Moreover, we provide applications of our findings by characterizing the directed completion of the Kruskal--Szekeres spacetime in terms of its radial null geodesics.

math.DG↗

arXiv2026

Closing the gap: Maz'ya-Shaposhnikova and asymptotics of fractional perimeters

We prove a generalization of the Maz'ya-Shaposhnikova formula in the case $p=2$ for functions that may not belong to ${L^2}(\mathbb{R}^d)$ and, thus, might not vanish at infinity. By introducing a notion of mass at infinity, we explicitly characterize the limit as $s\to0^+$ of Gagliardo seminorms localized on a bounded Lipschitz domain $\Omega$. By `localized', we mean here that we account only for interactions involving at least one point in $\Omega$. The identified limiting functional provides a unifying framework to link the classical Maz'ya-Shaposhnikova formula and the asymptotics of nonlocal perimeters. On the one hand, it reduces to the classical $L^2$ norm for functions that are globally integrable on $\mathbb{R}^d$. On the other hand, it recovers the pointwise limit of $s$-fractional perimeters when evaluated on characteristic functions of sets. We further show that the same functional encodes the asymptotic behavior of Gagliardo seminorms in the sense of Gamma-convergence with respect to the weak-$L^2$ topology. Finally, we provide an extension to the setting of metric measure spaces.

math.AP↗

arXiv2026

Existence and regularity of solutions to parabolic-elliptic nonlinear systems

In this paper we study the existence and summability of the solutions to the following parabolic-elliptic system of partial differential equations with discontinuous coefficients: \begin{equation*} \begin{cases} u_t - \operatorname{div}(A(x, t) \nabla u) = -\operatorname{div}(u M(x) \nabla \psi) + f(x, t) & \text{in } \Omega_T, \\ -\operatorname{div}(M(x) \nabla \psi) = |u|^\theta & \text{in } \Omega_T, \\ \psi(x, t) = 0 & \text{on } \partial \Omega \times (0, T), \\ u(x, t) = 0 & \text{on } \partial \Omega \times (0, T), \\ u(x, 0) = 0 & \text{in } \Omega. \end{cases} \end{equation*} Here, $\Omega$ is an open and bounded subset of $\mathbb R^N$, $N>2$, $\theta\in(0,\frac{2}{N})$, $0 1$ and $q>1$.

math.AP↗

arXiv2025

Existence and summability of solutions to nonlinear X-elliptic equations with measurable coefficients

We prove an existence result for solutions to a class of nonlinear degenerate-elliptic equations with measurable coefficients and zero Dirichlet boundary condition. The main term is given by a nonlinear operator in divergence form associated to a family of vector fields which satisfy a Poincar\'e inequality and the doubling condition. Furthermore, we prove that the solutions satisfy a generalization of the $L^p$-regularity results which hold for the solutions to Leray-Lions type equations.

math.AP↗

arXiv2025

Existence, uniqueness, regularity and stability of solutions to linear $X$-elliptic equations with measurable coefficients

We prove an existence and uniqueness result for solutions to linear $X$-elliptic equations with $L^1$ data and zero Dirichlet boundary conditions. Such solutions depend continuously on the datum. Moreover, we show that an improvement in the summability of the data yields a corresponding improvement in the summability of the solutions, in a manner analogous to the one that occurs in the case of uniformly elliptic equations.

math.AP↗

arXiv2025

Mosco-convergence of convex sets and unilateral problems for differential operators with lower order terms having natural growth

We study the stability of solutions to a class of variational inequalities posed on obstacle-type convex sets, under Mosco-convergence. More precisely, for a fixed obstacle $\psi\in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega)$, we consider $u\in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega)$ satisfying $u\geq\psi$ a.e. and $$ \langle A(u),v-u\rangle+\int_{\Omega}H(x,u,\nabla u)(v-u)\geq 0$$ for all $v\in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega)$ with $v\geq\psi$. Here, $A$ is a Leray-Lions type operator, mapping $W_0^{1,p}(\Omega)$ into its dual $W^{-1, p'}(\Omega)$, while $H(x, u, D u)$ grows like $|D u|^p$. Our main result establishes that the solutions are stable under Mosco-convergence of the constraint sets. This extends classical stability results to natural growth problems.

math.AP↗

arXiv2025

Analysis for non-local phase transitions close to the critical exponent $s=\frac12$

We analyze the behaviour of double-well energies perturbed by fractional Gagliardo squared seminorms in $H^s$ close to the critical exponent $s=\frac12$. This is done by computing a scaling factor $\lambda(\varepsilon,s)$, continuous in both variables, such that \[ \mathcal{F}^{s_\varepsilon}_\varepsilon(u)=\frac{\lambda(\varepsilon,s_\varepsilon)}{\varepsilon}\int W(u)dt+\lambda(\varepsilon,s_\varepsilon)\varepsilon^{(2s_\varepsilon-1)^+}[u]_{H^{s_\varepsilon}}^2 \] $\Gamma$-converge, for any choice of $s_\varepsilon \to \frac12$ as $\varepsilon\to 0$, to the sharp-interface functional found by Alberti, Bouchitt\'e and Seppecher with the scaling ${|\log\varepsilon|^{-1}}$. Moreover, we prove that all the values $s\in [\frac12,1 )$ are regular points for the functional $\mathcal{F}^{s}_\varepsilon$ in the sense of equivalence by $\Gamma$-convergence introduced by Braides and Truskinovsky, and that the $\Gamma$-limits as $\varepsilon\to 0$ are continuous with respect to $s$. In particular, the corresponding surface tensions, given by suitable non-local optimal-profile problems, are continuous on $[\frac12,1)$.

math.AP↗