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Antonia Passarelli di Napoli

Publications and source records attributed to Antonia Passarelli di Napoli.

At least 19 recordsLinked to original sources
arXiv2026

Regularity for Minimizers of non Autonomous Singular Functionals with Anisotropic Growth

We establish the higher differentiability of the local minimizers to a class of non autonomous convex integral functionals satisfying anisotropic subquadratic growth conditions, that include, as a particular case, those with orthotropic structure. The result is obtained under a gap bound on the exponents \(p_i\), that guarantees the local boundedness of the minimizers and under a suitable Sobolev assumption on the map that measures the oscillation of the energy density with respect to the $x$ variable, that is independent on the dimension.

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arXiv2026

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

The paper deals with the second order regularity properties of the weak solutions $u\in W^{1,ϕ}(Ω, \real^n)$ } of systems of the form \begin{equation*}\label{equareg} -\dive A(x,\E u)=f, \end{equation*} in a bounded domain $Ω\subset\R^n$, $n>2$, 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 $Φ$. 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.

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arXiv2026

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 Poincaré-Sobolev inequality in Orlicz spaces, in modular form, on domains with finite measure plays a role in the proofs.

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arXiv2026

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

We investigate the local boundedness of solutions $u:Ω_T\to\mathbb{R}$ to parabolic equations of the form \begin{equation*} \partial_tu-\mathrm{div}\,\mathcal{A}(x,t,Du)=0 \qquad\mbox{in }Ω_T=Ω\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,ξ)|&\le b(x,t)(μ^2+|ξ|^2)^{\frac{q-1}{2}},\\ \langle \mathcal{A}(x,t,ξ),ξ\rangle&\ge a(x,t)(μ^2+|ξ|^2)^{\frac {p-2}{2}}|ξ|^2, \end{align*} for $2\le p\le q$ and $μ\in[0,1]$, where the functions $a^{-1}, b:Ω_T\to\mathbb{R}$ 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.

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arXiv2025

On the second-order regularity of solutions to widely singular or degenerate elliptic equations

We consider local weak solutions to PDEs of the type \[ -\,\mathrm{div}\left((\vert Du\vert-λ)_{+}^{p-1}\frac{Du}{\vert Du\vert}\right)=f\,\,\,\,\,\,\,\text{in}\,\,Ω, \] where $1 2$, and that $f\in L_{loc}^{{\frac{np}{n(p-1)+2-p}}}(Ω)$ if $1<p\leq2$. The conditions on the datum $f$ are essentially sharp. As a consequence, we obtain the local higher integrability of $Du$ under the same minimal assumptions on $f$. For $λ=0$, our results give back those contained in [12,28].

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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.

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arXiv2023

Regularity results for a class of widely degenerate parabolic equations

Motivated by applications to gas filtration problems, we study the regularity of weak solutions to the strongly degenerate parabolic PDE $u_{t}-\mathrm{div}\left((\vert Du\vert-ν)_{+}^{p-1}\frac{Du}{\vert Du\vert}\right)=f$ in $Ω_{T}=Ω\times(0,T)$, where $Ω$ is a bounded domain in $\mathbb{R}^{n}$ for $n\geq2$, $p\geq2$, $ν$ is a positive constant and $\left(\,\cdot\,\right)_{+}$ stands for the positive part. Assuming that the datum $f$ belongs to a suitable Lebesgue-Sobolev parabolic space, we establish the Sobolev spatial regularity of a nonlinear function of the spatial gradient of the weak solutions, which in turn implies the existence of the weak time derivative $u_{t}$. The main novelty here is that the structure function of the above equation satisfies standard growth and ellipticity conditions only outside a ball with radius $ν$ centered at the origin. We would like to point out that the first result obtained here can be considered, on the one hand, as the parabolic counterpart of an elliptic result established in [5], and on the other hand as the extension to a strongly degenerate context of some known results for less degenerate parabolic equations.

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arXiv2023

Higher regularity for weak solutions to degenerate parabolic problems

In this paper, we study the regularity of weak solutions to the following strongly degenerate parabolic equation \begin{equation*} u_t-÷\left(\left(\left|Du\right|-1\right)_+^{p-1}\frac{Du}{\left|Du\right|}\right)=f\qquad\mbox{ in }Ω_T, \end{equation*} where $Ω$ is a bounded domain in $\mathbb{R}^{n}$ for $n\geq2$, $p\geq2$ and $\left(\,\cdot\,\right)_{+}$ stands for the positive part. We prove the higher differentiability of a nonlinear function of the spatial gradient of the weak solutions, assuming only that $f\in L^{2}_{\loc}\left(Ω_T\right)$. This allows us to establish the higher integrability of the spatial gradient under the same minimal requirement on the datum $f$.

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arXiv2022

Lipschitz regularity of minimizers of variational integrals with variable exponents

In this paper we prove the Lipschitz regularity for local minimizers of convex variational integrals of the form \[ \mathfrak{F}( v, Ω)= \int_Ω \! F(x, Dv(x)) \, dx, \] where, for ${n > 2}$ and $N\ge 1$, $Ω$ is a bounded open set in $\mathbb{R}^n$, $u \in W^{1,1}(Ω, \mathbb{R}^N)$ and the energy density $F:Ω\times \mathbb{R}^{N \times n}\to \mathbb{R}$ satisfies the so called variable growth conditions. The main novelty of the paper is that we assume an almost critical regularity in the Orlicz Sobolev setting for the energy density as a function of the $x$ variable.

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arXiv2022

Higher differentiability results for solutions to a class of non-homogeneouns elliptic problems under sub-quadratic growth conditions

We prove a sharp higher differentiability result for local minimizers of functionals of the form $$\mathcal{F}\left(w,Ω\right)=\int_Ω\left[ F\left(x,Dw(x)\right)-f(x)\cdot w(x)\right]dx$$ with non-autonomous integrand $F(x,ξ)$ which is convex with respect to the gradient variable, under $p$-growth conditions, with $1<p<2$. The main novelty here is that the results are obtained assuming that the partial map $x\mapsto D_ξF(x,ξ)$ has weak derivatives in some Lebesgue space $L^q$ and the datum $f$ is assumed to belong to a suitable Lebesgue space $L^r$. We also prove that it is possible to weaken the assumption on the datum $f$ and on the map $x\mapsto D_ξF(x,ξ)$, if the minimizers are assumed to be a priori bounded.

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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.

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arXiv2020

On the validity of variational inequalities for obstacle problems with non-standard growth

The aim of the paper is to show that the solutions to variational problems with non-standard growth conditions satisfy a corresponding variational inequality without any smallness assumptions on the gap between growth and coercitivity exponents. Our results rely on techniques based on Convex Analysis that consist in establishing duality formulas and pointwise relations between minimizers and corresponding dual maximizers, for suitable approximating problems, that are preserved passing to the limit. In this respect we are able to show that the right class of competitors are the functions with finite energy, in agreement with the unconstrained results.

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arXiv2020

Elliptic Equations With Degenerate weights

We obtain new local Calderon-Zygmund estimates for elliptic equations with matrix-valued weights for linear as well as non-linear equations. We introduce a novel log-BMO condition on the weight M. In particular, we assume smallness of the logarithm of the matrix-valued weight in BMO. This allows to include degenerate, discontinuous weights. We provide examples that show the sharpness of the estimates in terms of the log-BMO-norm.

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arXiv2019

Regularity results for a class of obstacle problems with $p,q-$growth conditions

In this paper we prove the local boundedness as well as the local Lipschitz continuity for solutions to a class of obstacle problems of the type $$\min\left\{\int_Ω{F(x, Dz)}: z\in \mathcal{K}_ψ(Ω)\right\}.$$ Here $\mathcal{K}_ψ(Ω)$ is set of admissible functions $z \in W^{1,p}(Ω)$ such that $z \ge ψ$ a.e. in $Ω$, $ψ$ being the obstacle and $Ω$ being an open bounded set of $\mathbb{R}^n$, $n \ge 2$. The main novelty here is that we are assuming $ F(x, Dz)$ satisfying $(p,q)$-growth conditions {and less restrictive assumptions on the obstacle with respect to the existing regularity results}.

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arXiv2019

Very degenerate elliptic equations under almost critical Sobolev regularity

We prove the local Lipschitz continuity and the higher differentiability of local minimizers of integral functionals with non autonomous integrand which is degenerate convex with respect to the gradient variable. The main novelty here is that the results are obtained assuming that the coefficients have weak derivative in an almost critical Zygmund class and the datum f is assumed to belong to the same Zygmund class.

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arXiv2016

Fractional differentiability for solutions of nonlinear elliptic equations

We study nonlinear elliptic equations in divergence form $${\operatorname{div}}{\mathcal A}(x,Du)={\operatorname{div}}G.$$ When ${\mathcal A}$ has linear growth in $Du$, and assuming that $x\mapsto{\mathcal A}(x,ξ)$ enjoys $B^α_{\frac{n}α, q}$ smoothness, local well-posedness is found in $B^α_{p,q}$ for certain values of $p\in[2,\frac{n}α)$ and $q\in[1,\infty]$. In the particular case ${\mathcal A}(x,ξ)=A(x)ξ$, $G=0$ and $A\in B^α_{\frac{n}α,q}$, $1\leq q\leq\infty$, we obtain $Du\in B^α_{p,q}$ for each $p<\frac{n}α$. Our main tool in the proof is a more general result, that holds also if ${\mathcal A}$ has growth $s-1$ in $Du$, $2\leq s\leq n$, and asserts local well-posedness in $L^q$ for each $q>s$, provided that $x\mapsto{\mathcal A}(x,ξ)$ satisfies a locally uniform $VMO$ condition.

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arXiv2015

Regularity Results for an Optimal Design Problem with a Volume Constraint

Regularity results for minimal configurations of variational problems involving both bulk and surface energies and subject to a volume constraint are established. The bulk energies are convex functions with p-power growth, but are otherwise not subjected to any further structure conditions. For a minimal configuration (u;E), Hoelder continuity of the function u is proved as well as partial regularity of the boundary of the minimal set E. Moreover, full regularity of the boundary of the minimal set is obtained under suitable closeness assumptions on the eigenvalues of the bulk energies.

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