Higher Order Elliptic Equations on Nonsmooth Domains
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909491864272896 |
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| author | Geng, Jun |
| author_facet | Geng, Jun |
| contents | In 1995, D. Jerison and C. Kenig in \cite{JK-1995} considered the the inhomogeneous Dirichlet problem $Δu= f$ on $Ω$, $u=0$ on $\partialΩ$ in Lipschitz domains. One of their main results shows that the $W^{1,p}$ estimate holds for the sharp range $\frac{3}{2}-\varepsilon<p<3+\varepsilon$ for $d\geq 3$ and $\frac{4}{3}-\varepsilon<p<4+\varepsilon$ if $d=2$. Although the argument employed in \cite{JK-1995} yields optimal results, they rely on an essential fashion on the maximum principle and, as such, do not readily adapt to higher-order case. By using a new method, the aim of this paper is to establish an extension of their theorem for higher order inhomogeneous elliptic equations on bounded Lipschitz and convex domains, uniform $W^{\ell,p}$ estimates are obtained for $p$ in certain ranges. Especially, compare to the result in \cite{MM-2013} for biharmonic equation, a larger, sharp, range of $p's$ was obtained in this paper. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_09339 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Higher Order Elliptic Equations on Nonsmooth Domains Geng, Jun Analysis of PDEs In 1995, D. Jerison and C. Kenig in \cite{JK-1995} considered the the inhomogeneous Dirichlet problem $Δu= f$ on $Ω$, $u=0$ on $\partialΩ$ in Lipschitz domains. One of their main results shows that the $W^{1,p}$ estimate holds for the sharp range $\frac{3}{2}-\varepsilon<p<3+\varepsilon$ for $d\geq 3$ and $\frac{4}{3}-\varepsilon<p<4+\varepsilon$ if $d=2$. Although the argument employed in \cite{JK-1995} yields optimal results, they rely on an essential fashion on the maximum principle and, as such, do not readily adapt to higher-order case. By using a new method, the aim of this paper is to establish an extension of their theorem for higher order inhomogeneous elliptic equations on bounded Lipschitz and convex domains, uniform $W^{\ell,p}$ estimates are obtained for $p$ in certain ranges. Especially, compare to the result in \cite{MM-2013} for biharmonic equation, a larger, sharp, range of $p's$ was obtained in this paper. |
| title | Higher Order Elliptic Equations on Nonsmooth Domains |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2502.09339 |