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Elvira Mascolo

Publications and source records attributed to Elvira Mascolo.

9 recordsLinked to original sources
arXiv2026

Local Boundedness of Local Minimizers for a Class of Nonlinear Elliptic Systems with General Growth

In this paper, we prove the local boundedness of solutions to systems of partial differential equations in divergence form. More specifically, we consider systems that include the first variations of functionals depending on the spatial variable and exhibiting nonstandard growth with respect to the gradient, such as $$\int_Ω \left( 1+ h(|Du|)\right) ^{α(x)} \, d x,$$ where the convex function $h=h(t)$ does not satisfy the so-called $Δ_2$ property and does not exhibit the conventional polynomial growth behavior.

math.AP↗
arXiv2026

Regularity of vectorial minimizers for non-uniformly elliptic anisotropic integrals

We establish the local boundedness of the local minimizers $u:Ω\rightarrow\mathbb{R}^{m}$ of non-uniformly elliptic integrals of the form $\int_Ωf(x,Dv)\,dx$, where $Ω$ is a bounded open subset of $\mathbb{R}^{n}$ ($n\geq2)$ and the integrand satisfies anisotropic growth conditions of the type \[ \sum_{i=1}^{n}λ_{i}(x)|ξ_{i}|^{p_{i}}\le f(x,ξ)\leμ(x)\left\{ 1+|ξ|^{q}\right\} \] for some exponents $q\geq p_{i}>1$ and with non-negative functions $λ_{i},μ$ fulfilling suitable summability assumptions. The main novelties here are the degenerate and anisotropic behaviour of the integrand and the fact that we also address the case of vectorial minimizers ($m>1$). Our proof is based on the celebrated Moser iteration technique and employs an embedding result for anisotropic Sobolev spaces.

math.AP↗
arXiv2025

Local boundedness for solutions of a class of non-uniformly elliptic anisotropic problems

We consider a class of {energy integrals}, associated to nonlinear and non-uniformly elliptic equations, with integrands $f(x,u,ξ)$ satisfying anisotropic $p_i,q$-growth conditions of the form $$ \sum_{i=1}^n λ_i (x)|ξ_i|^{p_i}\le {f}(x,u,ξ)\le μ(x)\left\{|ξ|^{q} + |u|^γ+1\right\} $$ for some exponents $γ\ge q\geq p_i>1$, and non-negative functions $λ_i,μ$ subject to suitable summability assumptions. We prove the local boundedness of scalar local quasi-minimizers of such integrals.

math.AP↗
arXiv2024

Regularity for minimizers of scalar integral functionals

We prove the local Lipschitz regularity of the local minimizers of scalar integral functionals of the form \begin{equation*} \mathcal{F}(v;Ω)= \int_Ω f (x, Dv) dx \end{equation*} under $(p,q)$-growth conditions. The main novelty is that, beside a suitable regularity assumption on the partial map $x\mapsto f(x,ξ)$, we do not assume any special structure for the energy density as a function of the $ξ$-variable.

math.AP↗
arXiv2023

Regularity for nonuniformly elliptic equations with $p,q-$growth and explicit $x,u-$dependence

We are interested in the regularity of weak solutions $u$ to the elliptic equation in divergence form; precisely in their local boundedness and their local Lipschitz continuity under general growth conditions, the so called $p,q-$growth conditions. We found a unique set of assumptions to get all these regularity properties at the same time; in the meantime we also found the way to treat a more general context, with explicit dependence on $( x,u) $, other than on the gradient variable $ξ=Du$; these aspects require particular attention due to the $p,q-$context, with some differences and new difficulties compared to the standard case $p=q$.

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arXiv2023

The Leray-Lions existence theorem under general growth conditions

We prove an existence result of weak solutions $u\in W_{0}^{1,p}\left( Ω\right) \cap W_{\mathrm{loc}}^{1,q}\left( Ω\right) $, to a Dirichlet problem for a second order elliptic equation in divergence form, under general and $p,q-$growth conditions of the differential operator. This is a first attempt to extend to general growth the well known Leray-Lions existence theorem, which holds under the so-called natural growth conditions with $q=p$. We found a way to treat the general context with explicit dependence on $\left( x,u\right) $, other than on the gradient variable $ξ=Du$; these aspects require particular attention due to the $p,q-$context, with some differences and new difficulties compared to the standard case $p=q$.

math.AP↗
arXiv2021

Higher differentiability for a class of problems under p,q subquadratic growth

We study the higher differentiability for nonlinear elliptic equation in divergence form $\mathcal{A}(x,Du)=b(x)$. The result covers the cases in which $\mathcal{A}(x, ξ)$ satisfies $p,q$ growth, with $1<p<2$ in $ξ$ and a Sobolev dependence of with respect to $x$. By means of an a-priori estimate we ensure the $W^{2,p}_{\mathrm{loc}}(Ω)$-property for the solution of the boundary value problem.

math.AP↗
arXiv2021

Lipschitz regularity for degenerate elliptic integrals with p,q-growth

We establish the local Lipschitz continuity and the higher differentiability of vector-valued local minimizers of a class of energy integrals of the Calculus of Variations. The main novelty is that we deal with possibly degenerate energy densities with respect to the x-variable.

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