arXiv subjects

Fabio Paronetto

Publications and source records attributed to Fabio Paronetto.

6 recordsLinked to original sources

arXiv2026

$H$-convergence and $\Gamma$-convergence in the Riesz fractional setting: the nonlinear case

This paper concerns the $H$-convergence of nonlinear nonlocal monotone operators defined through the Riesz fractional gradient and divergence. We show that the $H$-convergence in this nonlocal framework is equivalent to the $H$-convergence of the corresponding local one. As a consequence, we obtain a $H$-compactness result for a suitable class of nonlocal monotone operators. We then study the $\Gamma$-convergence of nonlocal energy functionals associated with the subclass of \emph{conservative} monotone operators, proving that it is equivalent to the $\Gamma$-convergence of the corresponding local energies. A key ingredient is a new uniqueness result for the integral representation of both local and nonlocal functionals. As a by-product, we obtain the $\Gamma$-compactness of the class of nonlocal energies under consideration. Finally, we show the equivalence between the $H$-convergence of nonlocal conservative monotone operators and the $\Gamma$-convergence of the associated energy functionals.

math.AP↗

arXiv2025

Variational convergences under moving anisotropies

We study the asymptotic behaviour of sequences of integral functionals depending on moving anisotropies. We introduce and describe the relevant functional setting, establishing uniform Meyers-Serrin type approximations, Poincaré inequalities and compactness properties. We prove several $Γ$-convergence results, and apply the latter to the study of $H$-convergence of anisotropic linear differential operators.

math.AP↗

arXiv2023

A Harnack inequality for solutions of elliptic-parabolic equations

We want to prove a Harnack type inequality for solutions of strongly degenerate parabolic, or elliptic-parabolic, equations. To do that, we first define a De Giorgi class of order $p = 2$ that contains the solutions of evolution equations of the types $\uprho (x,t) u_t + A u = 0$ and $(\uprho (x,t) u)_t + A u = 0$, where $\uprho > 0$ almost everywhere and $A$ is a suitable elliptic operator. For functions belonging to this class we prove an inhomogeneous parabolic Harnack inequality, i.e. a Harnack inequality that takes into account the mean value of $\uprho$ in different regions of $\Omega \times (0,T)$. \\ As a consequence, thanks to an approximation result and a delicate passage to the limit, we are able to get a Harnack inequality for solutions, and in these cases only for solutions, of strongly degenerating parabolic equations, i.e. when $\uprho \geqslant 0$. \\ As a byproduct one obtains H\"older continuity for solutions of a subclass of the first equation (i.e. $\uprho (x,t) u_t + A u = 0$): in particular the solutions of this subclass are H\"older continuous in the interface where $\uprho$ changes its sign, from positive to zero.

math.AP↗

arXiv2022

$G$-convergence of elliptic and parabolic operators depending on vector fields

We consider sequences of elliptic and parabolic operators in divergence form and depending on a family of vector fields. We show compactness results with respect to G-convergence, or H-convergence, by means of the compensated compactness theory, in a setting in which the existence of affine functions is not always guaranteed, due to the nature of the family of vector fields.

math.AP↗

arXiv2015

A Harnack's inequality for mixed type evolution equations

We define a homogeneous parabolic De Giorgi classes of order 2 which suits a mixed type class of evolution equations whose simplest example is $μ(x) \frac{\partial u}{\partial t} - Δu = 0$ where $μ$ can be positive, null and negative, so in particular elliptic-parabolic and forward-backward parabolic equations are included. For functions belonging to this class we prove local boundedness and show a Harnack inequality which, as by-products, gives Hölder-continuity, in particular in the interface $I$ where $μ$ change sign, and a maximum principle.

math.AP↗

arXiv2013

Local higher integrability for parabolic quasiminimizers in metric spaces

Using variational methods, we prove local higher integrability for the minimal p-weak upper gradients of parabolic quasiminimizers in metric measure spaces. We assume the measure to be doubling and the underlying space to be such that a weak Poincaré inequality is supported. We give proofs to density results concerning the space of test functions used when proving estimates for parabolic quasiminimizers.

math.AP↗