arXiv subjects

Flavia Giannetti

Publications and source records attributed to Flavia Giannetti.

8 recordsLinked to original sources

arXiv2026

Sobolev regularity of the symmetric gradient of solutions to a class of $\phi$-Laplacian systems

The paper deals with the second order regularity properties of the weak solutions $u\in W^{1,\phi}(\Omega, \real^n)$ } of systems of the form \begin{equation*}\label{equareg} -\dive A(x,\E u)=f, \end{equation*} in a bounded domain $\Omega\subset\R^n$, $n>2$, where the operator $ A(x,P)$ is Lipschitz continuous with respect to the $x$-variable and satisfies growth conditions with respect to the second variable expressed through a Young function $\Phi$. We prove the Sobolev regularity of a function of the symmetric gradient $\E u$ that takes into account the nonlinear growth of the operator $A(x,P)$, {assuming that the force term $f$ belongs to a suitable Orlicz-Sobolev space. {The main result is achieved through some uniform higher differentiability estimates for solutions to a class of approximating problems, constructed adding singular higher order perturbations to the system.

math.AP↗

arXiv2026

Local boundedness for solutions to parabolic $p,q$-problems with degenerate coefficients

We investigate the local boundedness of solutions $u:\Omega_T\to\mathbb{R}$ to parabolic equations of the form \begin{equation*} \partial_tu-\mathrm{div}\,\mathcal{A}(x,t,Du)=0 \qquad\mbox{in }\Omega_T=\Omega\times(0,T) \end{equation*} that satisfy $p,q$-growth conditions and have degenerate coefficients. More precisely, we assume structure conditions of the type \begin{align*} |\mathcal{A}(x,t,\xi)|&\le b(x,t)(\mu^2+|\xi|^2)^{\frac{q-1}{2}},\\ \langle \mathcal{A}(x,t,\xi),\xi\rangle&\ge a(x,t)(\mu^2+|\xi|^2)^{\frac {p-2}{2}}|\xi|^2, \end{align*} for $2\le p\le q$ and $\mu\in[0,1]$, where the functions $a^{-1}, b:\Omega_T\to\mathbb{R}$ are possibly unbounded and only satisfy some integrability condition. Under a certain assumption on the gap between $p$ and $q$, we prove two main results. First, we show that subsolutions that are contained in the natural energy space are locally bounded from above. Second, for parabolic equations with a variational structure, we use these bounds to show the existence of locally bounded variational solutions.

math.AP↗

arXiv2025

Fractional higher differentiability of solutions to strongly nonlinear Stokes systems

This work concerns stationary Stokes type systems governed by a general class of non-necessarily power-type nonlinearities. Fractional regularity properties of the symmetric gradient of local solutions are established, depending on a balance between the nonlinearity of the differential operator and the degree of integrability of the datum on right-hand side. The non-polynomial character of the differential operators calls for the use of Orlicz and Orlicz-Sobolev spaces as an appropriate functional framework for both the solutions and the datum. The regularity result amounts to the membership of a nonlinear expression of the symmetric gradient in Besov spaces. Fractional regularity of the pressure term is also exhibited and is formulated in terms of Orlicz-Besov spaces. Fractional Sobolev regularity of the symmetric gradient and of the pressure follow as a consequence.} Parallel results for the symmetric gradient of local solutions to the associated plain elliptic system are also offered. A new version of a Poincar\'e-Sobolev inequality in Orlicz spaces, in modular form, on domains with finite measure plays a role in the proofs.

math.AP↗

arXiv2025

Local Lipschitz continuity of the minimizers of nonuniformly convex functionals under the Lower Bounded Slope Condition

We prove the local Lipschitz regularity of the minimizers of functionals of the form \[ \mathcal I(u)=\int_\Omega f(\nabla u(x))+g(x)u(x)\,dx\qquad u\in\phi+W^{1,1}_0(\Omega) \] where $g$ is bounded and $\phi$ satisfies the Lower Bounded Slope Condition. The function $f$ is assumed to be convex but not uniformly convex everywhere. As byproduct, we also prove the existence of a locally Lipschitz minimizer for a class of functionals of the type above but allowing to the function $f$ to be nonconvex.

math.AP↗

arXiv2023

On the monotonicity of non-local perimeter of convex bodies

Under mild assumptions on the kernel $K\ge0$, the non-local $K$-perimeter $P_K$ satisfies the monotonicity property on nested convex bodies, i.e., if $A\subset B\subset\mathbb{R}^n$ are two convex bodies, then $P_K(A)\le P_K(B)$. In this note, we prove quantitative lower bounds on the difference of the $K$-perimeters of $A$ and $B$ in terms of their Hausdorff distance, provided that $K$ satisfies suitable symmetry properties.

math.MG↗

arXiv2023

On the convex components of a set in $\mathbb{R}^n$

We prove a lower bound on the number of the convex components of a compact set with non-empty interior in $\mathbb{R}^n$ for all $n\ge2$. Our result generalizes and improves the inequalities previously obtained in M. Carozza, F. Giannetti, F. Leonetti and A. Passarelli di Napoli, "Convex components", in Communications in Contemporary Mathematics, Vol. 21, No. 06, 1850036 (2019) and in M. La Civita and F. Leonetti, "Convex components of a set and the measure of its boundary", Atti. Sem. Mat. Fis. Univ. Modena Reggio Emilia 56 (2008-2009) 71-78.

math.MG↗

arXiv2020

Wolff potentials and local behaviour of solutions to measure data elliptic problems with Orlicz growth

We establish pointwise estimates expressed in terms of a nonlinear potential of a generalized Wolff type for $A$-superharmonic functions with nonlinear operator $A:Ω\times\mathbb{R}^n\to\mathbb{R}^n$ having measurable dependence on the spacial variable and Orlicz growth with respect to the last variable. The result is sharp as the same potential controls bounds from above and from below. Applying it we provide a bunch of precise regularity results including continuity and Hölder continuity for solutions to problems involving measures that satisfies conditions expressed in the natural scales. Finally, we give a variant of Hedberg--Wolff theorem on characterization of the dual of the Orlicz space.

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arXiv2018

Elliptic problems with growth in nonreflexive Orlicz spaces and with measure or $L^1$ data

We investigate solutions to nonlinear elliptic Dirichlet problems of the type \[ \left\{\begin{array}{cl} - {\rm div} A(x,u,\nabla u)= μ&\qquad \mathrm{ in}\qquad Ω, u=0 &\qquad \mathrm{ on}\qquad \partialΩ, \end{array}\right. \] where $Ω$ is a bounded Lipschitz domain in $\mathbb{R}^n$ and $A(x,z,ξ)$ is a Carathéodory's function. The growth of~the~monotone vector field $A$ with respect to the $(z,ξ)$ variables is expressed through some $N$-functions $B$ and $P$. We do not require any particular type of growth condition of such functions, so we deal with problems in nonreflexive spaces. When the problem involves measure data and weakly monotone operator, we prove existence. For $L^1$-data problems with strongly monotone operator we infer also uniqueness and regularity of~solutions and their gradients in the scale of Orlicz-Marcinkiewicz spaces.

math.AP↗