论文标题
在关键维度下，弱解决方案的弱解决方案的规律性
Regularity of weak solutions to higher order elliptic systems in critical dimensions
论文作者
论文摘要
在本文中，我们以Rivière和Struwe的精神（Comm。Pure。Appl。Math。2008）开发了一种基础和统一的处理，以探索在不使用保护法的情况下，在关键维度中探索高阶几何椭圆形系统的弱解决方案的规律性。结果，我们获得了de Longueville和Gastel和Gastel \ cite {Delongueville-Gastel-2019}的高阶椭圆系统的解决方案的内部连续性v_ {i}，du \ right \ rangle +\ sum_ {i = 0}^{k-2}Δ^{i}δ\ left（w_ {i} du \ right）\ quad \ quad \ quad \ quad \ text {in} b^{2k}，$$ unightions b^{2k}，$$ $ prionger of Coptictions coptions on Copeftions pableptions copertions on Coplitions puttion。这验证了Rivière的期望，并在$ k = 2 $时为Struwe的公开问题提供了肯定的答案。 Hölder的连续性也改善了Lamm和Rivière和De Longueville和Gastel的连续性结果。
In this paper, we develop an elementary and unified treatment, in the spirit of Rivière and Struwe (Comm. Pure. Appl. Math. 2008), to explore regularity of weak solutions of higher order geometric elliptic systems in critical dimensions without using conservation law. As a result, we obtain an interior Hölder continuity for solutions of the higher order elliptic system of de Longueville and Gastel \cite{deLongueville-Gastel-2019} in critical dimensions $$Δ^{k}u=\sum_{i=0}^{k-1}Δ^{i}\left\langle V_{i},du\right\rangle +\sum_{i=0}^{k-2}Δ^{i}δ\left(w_{i}du\right) \quad \text{in } B^{2k},$$ under critical regularity assumptions on the coefficient functions. This verifies an expectation of Rivière, and provides an affirmative answer to an open question of Struwe in dimension four when $k=2$. The Hölder continuity is also an improvement of the continuity result of Lamm and Rivière and de Longueville and Gastel.
