Unified a-priori estimates for minimizers under $p,q-$growth and exponential growth
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| Format: | Preprint |
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2024
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| _version_ | 1866913568895533056 |
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| author | Marcellini, Paolo Nastasi, Antonella Camacho, Cintia Pacchiano |
| author_facet | Marcellini, Paolo Nastasi, Antonella Camacho, Cintia Pacchiano |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_22875 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Unified a-priori estimates for minimizers under $p,q-$growth and exponential growth Marcellini, Paolo Nastasi, Antonella Camacho, Cintia Pacchiano Analysis of PDEs Functional Analysis 35D30, 35J15, 35J60, 49N60 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. |
| title | Unified a-priori estimates for minimizers under $p,q-$growth and exponential growth |
| topic | Analysis of PDEs Functional Analysis 35D30, 35J15, 35J60, 49N60 |
| url | https://arxiv.org/abs/2410.22875 |