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Paolo Marcellini

Publications and source records attributed to Paolo Marcellini.

7 recordsLinked to original sources
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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arXiv2024

Unified a-priori estimates for minimizers under $p,q-$growth and exponential growth

We propose some general growth conditions on the function $% f=f\left( x,ξ\right) $, including the so-called natural growth, or polynomial, or $p,q-$growth conditions, or even exponential growth, in order to obtain that any local minimizer of the energy integral $\;\int_{Ω}f\left( x,Du\right) dx\,$ is locally Lipschitz continuous in $Ω$. In fact this is the fundamental step for further regularity: the local boundedness of the gradient of any Lipschitz continuous local minimizer a-posteriori makes irrelevant the behavior of the integrand $f\left( x,ξ\right) $ as $\left\vert ξ\right\vert \rightarrow +\infty $; i.e., the general growth conditions a posteriori are reduced to a standard growth, with the possibility to apply the classical regularity theory. In other words, we reduce some classes of \textit{non-uniform} elliptic variational problems to a context of uniform ellipticity.

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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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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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arXiv2019

A-priori gradient bound for elliptic systems under either slow or fast growth conditions

We obtain an a-priori $W_{loc}^{1,\infty }\ ( Ω;\mathbb{R}^{m}\ ) -$bound for solutions in $Ω\subset \mathbb{R}^{n} $, $n\geq 2$, to the elliptic system \begin{equation*} \sum_{i=1}^{n}\frac{\partial }{\partial x_{i}}\ ( \frac{g_{t}\ ( x,\ |Du\ | \ ) }{\ |Du\ | } u_{x_{i}}^{α}\ ) =0,\;\;\;\;\;α=1,2,\ldots ,m, \end{equation*} where $g\ ( x,t\ ) $, $g:Ω\times \ [ 0,\infty \ ) \rightarrow \ [ 0,\infty \ ) $, is a Carathéodory function, convex and increasing with respect to the gradient variable $t\in \ [ 0,\infty \ ) $. We allow $x-$dependence, which turns out to be a relevant difference with respect to the autonomous case and not only a technical perturbation. Our assumptions allow us to consider both fast and slow growth. We allow fast growth even of exponential type; and slow growth, for instance of Orlicz-type with energy-integrands such as $g\ ( x,\ | Du\ |\ ) =|Du|\log (1+|Du|)$ or, when $n=2,3$, even asymptotic linear growth with energy integrands of the type \begin{equation*} g\ ( x,\ | Du\ | \ ) =\ | Du\ | -a\ ( x\ ) \sqrt{\ | Du\ | }\,. \end{equation*}

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