Article
| Title: | Regularizing effect of the interplay between coefficients in some noncoercive integral functionals (English) |
| Author: | Zhang, Aiping |
| Author: | Feng, Zesheng |
| Author: | Gao, Hongya |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 74 |
| Issue: | 3 |
| Year: | 2024 |
| Pages: | 915-925 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | We are interested in regularizing effect of the interplay between the coefficient of zero order term and the datum in some noncoercive integral functionals of the type $$ \mathcal {J} (v)= \int _\Omega j(x,v,\nabla v) {\rm d}x +\int _\Omega a(x) |v|^{2} {\rm d} x -\int _\Omega fv {\rm d}x, \quad v\in W^{1,2}_{0}(\Omega ), $$ where $\Omega \subset \mathbb R^N$, $j$ is a Carathéodory function such that $\xi \mapsto j(x,s,\xi )$ is convex, and there exist constants $ 0\le \tau <1$ and $M>0$ such that $$ \frac { |\xi |^{2}}{(1+|s|)^{\tau }}\leq j(x,s,\xi )\leq M|\xi |^2 $$ for almost all $x\in \Omega $, all $s\in \mathbb R$ and all $\xi \in \mathbb R^N$. We show that, even if $0<a(x)$ and $f(x)$ only belong to $L^{1}(\Omega )$, the interplay $$|f(x)|\leq 2 Qa(x) $$ implies the existence of a minimizer $u \in W_0^{1,2} (\Omega )$ which belongs to $L^{\infty }(\Omega )$. (English) |
| Keyword: | regularizing effect |
| Keyword: | interplay |
| Keyword: | minimizer |
| Keyword: | noncoercive integral functional |
| MSC: | 49J45 |
| DOI: | 10.21136/CMJ.2024.0216-24 |
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| Date available: | 2024-10-03T12:40:48Z |
| Last updated: | 2024-10-04 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152589 |
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