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Giovanni Cupini

Publications and source records attributed to Giovanni Cupini.

13 recordsLinked to original sources
arXiv2026

Widely degenerate anisotropic diffusion: local boundedness and semicontinuity

We investigate the regularity of local weak solutions to evolution equations of the form \[ \partial_{t}u\,=\,\sum_{i=1}^{n}\,\partial_{x_{i}}\left[a_{i}(x,t)\,(\vert\partial_{x_{i}}u\vert-\delta_{i})_{+}^{p_{i}-1}\,\frac{\partial_{x_{i}}u}{\vert\partial_{x_{i}}u\vert}\right]\,\,\,\,\,\,\,\,\,\,\mathrm{in}\,\,\,\Omega_{T}\,=\,\Omega\times(0,T)\,, \] where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with $n\geq2$, the coefficients $a_{i}$ are measurable and bounded, $p_{i}>1$ and $\delta_{i}\geq0$ are fixed parameters. Under suitable assumptions on the exponents $p_{i}$, we first show that the local boundedness of weak solutions follows from their membership in an appropriate non-homogeneous parabolic De Giorgi class. We then establish the existence of semicontinuous representatives for local weak sub(super)-solutions. Our analysis extends analogous results available for less degenerate operators and generalizes the local boundedness results obtained in [7] to fully anisotropic, widely degenerate parabolic PDEs with non-smooth coefficients depending additionally on the space-time variables $(x,t)$, whose growth is governed by a family of exponents $p_{i}$ rather than by a single exponent.

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arXiv2026

A spherical flatness index and a stability inequality for harmonic pseudospheres

We introduce a new flatness index for the boundary of an open subset $\Omega$ of $\mathbb{R}^n$, $n\ge 2$. This index provides a necessary condition for $\partial\Omega$ to be a harmonic pseudosphere and sufficient conditions for a harmonic pseudosphere to be a Euclidean sphere. These conditions will follow from a stability inequality formulated in terms of a harmonic invariant, the Kuran gap, recently introduced by the last two authors.

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arXiv2025

Global boundedness of weak solutions with finite energy to a general class of Dirichlet problems

As explained in detail in the prologue to this manuscript, boundedness of weak solutions for general classes of elliptic equations in divergence form is a classic tool for achieving higher regularity. We propose here some global boundedness results under general assumptions that can be applied to several cases studied in the recent and extensive literature on partial differential equations \textit{under general growth}. In particular, we propose the class of \textit{weak solutions with finite energy} in which to search for solutions and in which regularity can be studied and achieved. We emphasize that we are not limited to minimizers of certain integral functionals, as often considered recently in this context of general growth, but to the broader class of weak solutions to Dirichlet problems for general nonlinear elliptic equations in divergence form.

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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 $f(x,u,\xi)$ satisfying anisotropic $p_i,q$-growth conditions of the form $$ \sum_{i=1}^n \lambda_i (x)|\xi_i|^{p_i}\le {f}(x,u,\xi)\le \mu (x)\left\{|\xi|^{q} + |u|^{\gamma}+1\right\} $$ for some exponents $\gamma\ge q\geq p_i>1$, and non-negative functions $\lambda_i,\mu$ subject to suitable summability assumptions. We prove the local boundedness of scalar local quasi-minimizers of such integrals.

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arXiv2025

Regularity of vectorial minimizers for non-uniformly elliptic anisotropic integrals

We establish the local boundedness of the local minimizers $u:\Omega\rightarrow\mathbb{R}^{m}$ of non-uniformly elliptic integrals of the form $\int_{\Omega}f(x,Dv)\,dx$, where $\Omega$ is a bounded open subset of $\mathbb{R}^{n}$ ($n\geq2)$ and the integrand satisfies anisotropic growth conditions of the type \[ \sum_{i=1}^{n}\lambda_{i}(x)|\xi_{i}|^{p_{i}}\le f(x,\xi)\le\mu(x)\left\{ 1+|\xi|^{q}\right\} \] for some exponents $q\geq p_{i}>1$ and with non-negative functions $\lambda_{i},\mu$ 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.

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arXiv2024

A representation formula for regular functions on the characteristic plane of the second Heisenberg group

The aim of this paper is to study a Laplace-type operator and its fundamental solution on the characteristic plane in the Heisenberg group $\mathbb{H}^2$. We introduce a conformal version of the Laplacian and we prove that the distance induced by the immersion in the ambient space is a good approximation of its fundamental solution. We provide in particular a representation formula for smooth functions in terms of the gradient of the function and the gradient of the approximated fundamental solution. This representation formula in the plane is stable up to its characteristic point.

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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 $p=q$.

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arXiv2023

On the harmonic characterization of the spheres: a sharp stability inequality and some of its consequences

Let $ D$ be a bounded open subset of $\mathbb R^n$ with $|\partial D| < \infty$ and let $x_0 $ be a point of $D$. We introduce a new parameter, that we call Kuran gap of $\partial D$ w.r.t. $x_0$. Roughly speaking, this parameter, denoted by $\mathcal{K}(\partial D, x_0)$, measures the gap between $u(x_0)$ and the average of $u$ on $\partial D$ for a particular family of functions $u$ harmonic in $\overline{D}$, in terms of the Poisson kernel of the biggest ball $B$ centered at $x_0$ and contained in $D$. To do that, we need the domain $D$ Lyapunov-Dini regular in at least one of the points of $\partial D$ nearest to $x_0$. Our main stability result can be described as follows: $\mathcal{K}(\partial D, x_0)$ is bounded from below by a kind of isoperimetric index, precisely the normalized difference beetween $|\partial D|$ and $|\partial B|$. This extends a stability result by Preiss and Toro, and a more recent theorem by Agostiniani and Magnanini. Moreover, from our stability inequality we obtain a new sufficient condition for a harmonic pseudosphere to be a Euclidean sphere, a result which partially improves a rigidity theorem by Lewis and Vogel. Finally, we give a new solution of the surface version of a solid ``potato'' problem by Aharonov, Schiffer and Zalcman.

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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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arXiv2017

Schauder estimates at the boundary for sub-laplacians in Carnot groups

In this paper we prove Schauder estimates at the boundary for sub-Laplacian type operators in Carnot groups. While internal Schauder estimates have been deeply studied, up to now subriemannian estimates at the boundary are known only in the Heisenberg groups. The proof of these estimates in the Heisenberg setting, due to Jerison, is based on the Fourier transform technique and can not be repeated in general Lie groups. After the result of Jerison no new contribution to the boundary problem has been provided. In this paper we introduce a new approach, which allows to built a Poisson kernel starting from the fundamental solution, from which we deduce the Schauder estimates at non characteristic boundary points.

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arXiv2012

Global $L^{p}$ estimates for degenerate Ornstein-Uhlenbeck operators with variable coefficients

We consider a class of degenerate Ornstein-Uhlenbeck operators in $\mathbb{R}^{N}$, of the kind [\mathcal{A}\equiv\sum_{i,j=1}^{p_{0}}a_{ij}(x) \partial_{x_{i}x_{j}}^{2}+\sum_{i,j=1}^{N}b_{ij}x_{i}\partial_{x_{j}}%] where $(a_{ij})$ is symmetric uniformly positive definite on $\mathbb{R}^{p_{0}}$ ($p_{0}\leq N$), with uniformly continuous and bounded entries, and $(b_{ij})$ is a constant matrix such that the frozen operator $\mathcal{A}_{x_{0}}$ corresponding to $a_{ij}(x_{0})$ is hypoelliptic. For this class of operators we prove global $L^{p}$ estimates ($1<p<\infty$) of the kind:% [|\partial_{x_{i}x_{j}}^{2}u|_{L^{p}(\mathbb{R}% ^{N})}\leq c{|\mathcal{A}u|_{L^{p}(\mathbb{R}^{N})}+|u|_{L^{p}(\mathbb{R}% ^{N})}} for i,j=1,2,...,p_{0}.] We obtain the previous estimates as a byproduct of the following one, which is of interest in its own:% [|\partial_{x_{i}x_{j}}^{2}u|_{L^{p}(S_{T})}\leq c{|Lu|_{L^{p}(S_{T})}+|u|_{L^{p}(S_{T})}}] for any $u\in C_{0}^{\infty}(S_{T}),$ where $S_{T}$ is the strip $\mathbb{R}^{N}\times[-T,T]$, $T$ small, and $L$ is the Kolmogorov-Fokker-Planck operator% [L\equiv\sum_{i,j=1}^{p_{0}}a_{ij}(x,t) \partial_{x_{i}x_{j}}% ^{2}+\sum_{i,j=1}^{N}b_{ij}x_{i}\partial_{x_{j}}-\partial_{t}%] with uniformly continuous and bounded $a_{ij}$'s.

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