$L^p$-regularity for fourth order elliptic systems with antisymmetric potentials in higher dimensions
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2021
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866917801420128256 |
|---|---|
| author | Guo, Chang-Yu Wang, Changyou Xiang, Chang-Lin |
| author_facet | Guo, Chang-Yu Wang, Changyou Xiang, Chang-Lin |
| contents | We establish an optimal $L^p$-regularity theory for solutions to fourth order elliptic systems with antisymmetric potentials in all supercritical dimensions $n\ge 5$:
$$
Δ^2 u=Δ(D\cdot\nabla u)+div(E\cdot\nabla u)+(ΔΩ+G)\cdot\nabla u +f
\qquad \ {\rm{in}}\ B^n,
$$
where $Ω\in W^{1,2}(B^n, so_m)$ is antisymmetric and $f\in L^p(B^n)$, and $D, E, Ω, G$ satisfy the growth condition (GC-4), under the smallness condition of a critical scale invariant norm of $\nabla u$ and $\nabla^2 u$. This system was brought into lights from the study of regularity of (stationary) biharmonic maps between manifolds by Lamm-Rivière, Struwe, and Wang. In particular, our results improve Struwe's Hölder regularity theorem to any Hölder exponent $α\in (0,1)$ when $f\equiv 0$, and have applications to both approximate biharmonic maps and heat flow of biharmonic maps.
As a by-product of the techniques, we also extend the $L^p$-regularity theory of harmonic maps by Moser to Rivière-Struwe's second order elliptic systems with antisymmetric potentials under the growth condition (GC-2) in all dimensions, which confirms an expectation by Sharp. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_07227 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | $L^p$-regularity for fourth order elliptic systems with antisymmetric potentials in higher dimensions Guo, Chang-Yu Wang, Changyou Xiang, Chang-Lin Analysis of PDEs 35J48, 35G50, 35B65 We establish an optimal $L^p$-regularity theory for solutions to fourth order elliptic systems with antisymmetric potentials in all supercritical dimensions $n\ge 5$: $$ Δ^2 u=Δ(D\cdot\nabla u)+div(E\cdot\nabla u)+(ΔΩ+G)\cdot\nabla u +f \qquad \ {\rm{in}}\ B^n, $$ where $Ω\in W^{1,2}(B^n, so_m)$ is antisymmetric and $f\in L^p(B^n)$, and $D, E, Ω, G$ satisfy the growth condition (GC-4), under the smallness condition of a critical scale invariant norm of $\nabla u$ and $\nabla^2 u$. This system was brought into lights from the study of regularity of (stationary) biharmonic maps between manifolds by Lamm-Rivière, Struwe, and Wang. In particular, our results improve Struwe's Hölder regularity theorem to any Hölder exponent $α\in (0,1)$ when $f\equiv 0$, and have applications to both approximate biharmonic maps and heat flow of biharmonic maps. As a by-product of the techniques, we also extend the $L^p$-regularity theory of harmonic maps by Moser to Rivière-Struwe's second order elliptic systems with antisymmetric potentials under the growth condition (GC-2) in all dimensions, which confirms an expectation by Sharp. |
| title | $L^p$-regularity for fourth order elliptic systems with antisymmetric potentials in higher dimensions |
| topic | Analysis of PDEs 35J48, 35G50, 35B65 |
| url | https://arxiv.org/abs/2111.07227 |