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Antonio Giuseppe Grimaldi

Publications and source records attributed to Antonio Giuseppe Grimaldi.

14 recordsLinked to original sources
arXiv2026

Almost Lipschitz regularity for solutions of elliptic equations with discontinuous coefficients

We are interested in the local higher integrability of solutions to elliptic equations with linear growth of the form $$-\text{ div}A(x,Du)=f(x). $$ Under a Besov regularity on the partial map $x \mapsto A(x,ξ)$ and a summability condition on the datum $f$ in the scale of Lebesgue spaces, we prove that the solutions are almost Lipschitz continuous, i.e. their gradients belong locally to $L^q$, for any finite exponent $q$. In turn, solutions are locally $γ$-Hölder continuous, for every $γ\in (0,1)$. The difficulty arising from the lack of an explicit second variation for the problem is overcome by testing the equation with a function proportional to a power of the finite difference quotient of the solution. To the best of our knowledge, this technique is used in this context for the first time. We also provide an example showing the sharpness of our result in the scale of Lebesgue spaces.

math.AP↗
arXiv2025

Regularity for minimizers of degenerate, non-autonomous, orthotropic integral functionals

We prove the higher differentiability of integer order of locally bounded minimizers of integral functionals of the form \begin{equation*} \mathcal{F}(u,Ω):= \,\sum_{i=1}^{n} \dfrac{1}{p_i}\displaystyle \int_Ω\, a_i(x) \lvert u_{x_i} \rvert^{p_i} dx- \int_Ωω(x)u(x) dx, \end{equation*} where the exponents $ p_i \geq 2 $ and the coefficients $ a_i(x) $ satisfy a suitable Sobolev regularity. The main novelty consists in dealing with non-autonomous, anisotropic functionals, which depend also on the solution.

math.AP↗
arXiv2025

Some obstacle problems for partially hinged plates and related optimization issues

We study optimization problems for partially hinged rectangular plates, modeling bridge roadways, in the presence of real and artificial obstacles. Real obstacles represent structural constraints to avoid, while artificial ones are introduced to enhance stability. For the former, aiming to prevent collisions, we set up a worst-case optimization problem in which we minimize the amplitude of oscillations with respect to the density distribution; for the latter, aiming to improve the torsional stability, we minimize, with respect to the obstacles, the maximum of a gap function quantifying the displacement between the long edges of the plate. For both problems, existence results are provided, along with a discussion about qualitative properties of optimal density distributions and obstacles.

math.OC↗
arXiv2025

Weak comparison principle for widely degenerate elliptic equations

We prove a comparison principle for local weak solutions to a class of widely degenerate elliptic equations of the form \begin{equation} -\text{div} \left( \left(|Du|-1 \right)^{p-1}_+\frac{Du}{|Du|} \right) = f(x,u) \qquad \text{ in } Ω,\notag \end{equation} where $p \ge 2$ and $Ω$ is an open subset of $\mathbb{R}^{n}$, $n\geq2$. Moreover, we establish some second order regularity results of the solutions, that yields a weighted Sobolev inequality with widely degenerate weights.

math.AP↗
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].

math.AP↗
arXiv2024

Regularity results for minimizers of non-autonomous integral functionals

We establish the higher fractional differentiability for the minimizers of non-autonomous integral functionals of the form \begin{equation} \mathcal{F}(u,Ω):=\int_Ω\left[ f(x,Du)- g \cdot u \right] dx , \notag \end{equation} under $(p,q)$-growth conditions. Besides a suitable differentiability assumption on the partial map $x \mapsto D_ξf(x,ξ)$, we do not need to assume any differentiability assumption on the function $g$. Moreover, we show that the higher differentiability result holds true also assuming strict convexity and growth conditions on $f$ only at infinity.

math.AP↗
arXiv2024

Gradient regularity for a class of elliptic obstacle problems

We prove some regularity results for a priori bounded local minimizers of non-autonomous integral functionals of the form $$\mathcal{F}(v,Ω)=\int_ΩF(x,Dv)dx,$$ under the constraint $v \ge ψ$ a.e. in $Ω$, where $ψ$ is a fixed obstacle function. Assuming that the coefficients of the partial map $x \mapsto D_ξF(x,ξ)$ satisfy a suitable Sobolev regularity, we are able to obtain higher differentiability and Lipschitz continuity results for the local minimizers.

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

Higher regularity for minimizers of very degenerate convex integrals

In this paper, we consider minimizers of integral functionals of the type \begin{equation*} \mathcal{F}(u):= \int_Ω\dfrac{1}{p} \bigl( |Du(x)|_{γ(x)}-1\bigr)_+^p \ \mathrm{d}x, \end{equation*} for $p >1$, where $u : Ω\subset \mathbb{R}^n \to \mathbb{R}^N$, with $N \ge 1$, is a possibly vector-valued function. Here, $| \cdot |_γ$ is the associated norm of a bounded, symmetric and coercive bilinear form on $\mathbb{R}^{nN}$. We prove that $\mathcal{K}(x,Du)$ is continuous in $Ω$, for any continuous function $\mathcal{K}: Ω\times \mathbb{R}^{nN} \rightarrow \mathbb{R}$ vanishing on $\bigl\{ (x,ξ) \in Ω\times \mathbb{R}^{nN} : |ξ|_{γ(x)} \le 1 \bigr\}$.

math.AP↗
arXiv2022

Higher differentiability for bounded solutions to a class of obstacle problems with $(p,q)$-growth

We establish the higher fractional differentiability of bounded minimizers to a class of obstacle problems with non-standard growth conditions of the form \begin{gather*} \min \biggl\{ \displaystyle\int_Ω F(x,Dw)dx \ : \ w \in \mathcal{K}_ψ(Ω) \biggr\}, \end{gather*} where $Ω$ is a bounded open set of $\mathbb{R}^n$, $n \geq 2$, the function $ψ\in W^{1,p}(Ω)$ is a fixed function called \textit{obstacle} and $\mathcal{K}_ψ(Ω) := \{ w \in W^{1,p}(Ω) : w \geq ψ\ \text{a.e. in} \ Ω\}$ is the class of admissible functions. If the obstacle $ψ$ is locally bounded, we prove that the gradient of solution inherits some fractional differentiability property, assuming that both the gradient of the obstacle and the mapping $x \mapsto D_ξF(x,ξ)$ belong to some suitable Besov space. The main novelty is that such assumptions are not related to the dimension $n$.

math.AP↗
arXiv2022

Higher differentiability results in the scale of Besov spaces to a class of double-phase obstacle problems

We study the higher fractional differentiability properties of the gradient of the solutions to variational obstacle problems of the form \begin{gather*} \min \biggl\{ \int_Ω F(x,w,Dw) d x \ : \ w \in \mathcal{K}_ψ(Ω) \biggr\}, \end{gather*} with $F$ double phase functional of the form \begin{equation*} F(x,w,z)=b(x,w)(|z|^p+a(x)|z|^q), \end{equation*} where $Ω$ is a bounded open subset of $\mathbb{R}^n$, $ψ\in W^{1,p}(Ω)$ is a fixed function called \textit{obstacle} and $\mathcal{K}_ψ(Ω)= \{ w \in W^{1,p}(Ω) : w \geq ψ\ \text{a.e. in} \ Ω\}$ is the class of admissible functions. Assuming that the gradient of the obstacle belongs to a suitable Besov space, we are able to prove that the gradient of the solution preserves some fractional differentiability property.

math.AP↗
arXiv2021

Higher fractional differentiability for solutions to a class of obstacle problems with non-standard growth conditions

We here establish the higher fractional differentiability for solutions to a class of obstacle problems with non-standard growth conditions. We deal with the case in which the solutions to the obstacle problems satisfy a variational inequality of the form \begin{equation*} \displaystyle\int_Ω \langle \mathcal{A}(x,Du) ,D(φ-u) \rangle dx \geq 0 \qquad \forall φ\in \mathcal{K}_ψ(Ω), \end{equation*} where $Ω$ is a bounded open subset of $\mathbb{R}^n$, $ψ\in W^{1,p}(Ω)$ is a fixed function called \textit{obstacle} and $\mathcal{K}_ψ(Ω)= \{ w \in W^{1,p}(Ω) : w \geq ψ\ \text{a.e. in} \ Ω\}$ is the class of admissible functions. Assuming that the gradient of the obstacle belongs to some suitable Besov space, we are able to prove that some fractional differentiability property transfers to the gradient of the solution.

math.AP↗