On the interplay between $(p,q)$-growth and $x$-dependence of the energy integrand: a limit case
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917270730571776 |
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| author | Eleuteri, M. Marcellini, P. Mascolo, E. di Napoli, A. Passarelli |
| author_facet | Eleuteri, M. Marcellini, P. Mascolo, E. di Napoli, A. Passarelli |
| contents | We establish the local Lipschitz regularity of the local minimizers of non autonomous integral funtionals of the form \[ \int_ΩF(x, Dz)\,dx, \] where $Ω$ is a bounded open set of $\mathbb{R}^n$, $n \ge 2$. The energy density $F(x,ξ)$ satisfies $(p,q)-$growth conditions with respect to the gradient variable and belongs to the Sobolev class $W^{1,ϕ}$, with $ϕ(t)=t^r\log^α(e+t),$ $r\ge n$, $α\ge 0$, as a function of the $x$ variable, under the condition $$ 1\le\frac{q}{p} \le 1 + \frac{1}{n} - \frac{1}{r}. $$ We present a unified approach that covers the limit case $$ \frac{q}{p} = 1 + \frac{1}{n} - \frac{1}{r} $$ and retrieves the results in \cite{EMM16} and in \cite{CGHPdN20}. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_12033 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the interplay between $(p,q)$-growth and $x$-dependence of the energy integrand: a limit case Eleuteri, M. Marcellini, P. Mascolo, E. di Napoli, A. Passarelli Analysis of PDEs 35J87, 49J40, 47J20 We establish the local Lipschitz regularity of the local minimizers of non autonomous integral funtionals of the form \[ \int_ΩF(x, Dz)\,dx, \] where $Ω$ is a bounded open set of $\mathbb{R}^n$, $n \ge 2$. The energy density $F(x,ξ)$ satisfies $(p,q)-$growth conditions with respect to the gradient variable and belongs to the Sobolev class $W^{1,ϕ}$, with $ϕ(t)=t^r\log^α(e+t),$ $r\ge n$, $α\ge 0$, as a function of the $x$ variable, under the condition $$ 1\le\frac{q}{p} \le 1 + \frac{1}{n} - \frac{1}{r}. $$ We present a unified approach that covers the limit case $$ \frac{q}{p} = 1 + \frac{1}{n} - \frac{1}{r} $$ and retrieves the results in \cite{EMM16} and in \cite{CGHPdN20}. |
| title | On the interplay between $(p,q)$-growth and $x$-dependence of the energy integrand: a limit case |
| topic | Analysis of PDEs 35J87, 49J40, 47J20 |
| url | https://arxiv.org/abs/2602.12033 |