Failure of $L^r$-Calderón-Zygmund estimates for the p-Laplace equation for small $r$
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913460454948864 |
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| author | Schikorra, Armin |
| author_facet | Schikorra, Armin |
| contents | Let $p \neq 2$. For any small enough $r> \max \{p-1,1\}$ and for any $Λ> 1$ there exists a Lipschitz function $u$ and a bounded vectorfield $f$ such that \[ \begin{cases}
{\rm div}(|\nabla u|^{p-2} \nabla u) = {\rm div} (f) \quad& \text{in $\mathbb{B}^2$}\\ u=0 &\text{on $\partial \mathbb{B}^2$} \end{cases}
\] but \[
\int_{\mathbb{B}^2} |\nabla u|^r \not \leq Λ\int_{\mathbb{B}^2} |f|^{\frac{r}{p-1}}. \] This disproves a conjecture by Iwaniec from 1983. The proof adapts recent convex-integration ideas by Colombo-Tione. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_03546 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Failure of $L^r$-Calderón-Zygmund estimates for the p-Laplace equation for small $r$ Schikorra, Armin Analysis of PDEs Let $p \neq 2$. For any small enough $r> \max \{p-1,1\}$ and for any $Λ> 1$ there exists a Lipschitz function $u$ and a bounded vectorfield $f$ such that \[ \begin{cases} {\rm div}(|\nabla u|^{p-2} \nabla u) = {\rm div} (f) \quad& \text{in $\mathbb{B}^2$}\\ u=0 &\text{on $\partial \mathbb{B}^2$} \end{cases} \] but \[ \int_{\mathbb{B}^2} |\nabla u|^r \not \leq Λ\int_{\mathbb{B}^2} |f|^{\frac{r}{p-1}}. \] This disproves a conjecture by Iwaniec from 1983. The proof adapts recent convex-integration ideas by Colombo-Tione. |
| title | Failure of $L^r$-Calderón-Zygmund estimates for the p-Laplace equation for small $r$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2408.03546 |