AI 中文总结
针对有界Lipschitz区域内散度方程研究零散、问题未解的现状，提出统一方法，找到满足椭圆正则性的特殊解，简化Bogovskii常数定义并证明其可达性，得到通用下界与非极小性判据，分析典型区域并引入高阶常数引出多调和Stokes问题，提出三个开放问题。
AI 中文摘要
关于$\n\bR^n$（$n\bge2$）中有界Lipschitz区域内的散度方程，已有大量文献，但内容相当零散，即便看似简单的问题也仍未得到解决。我们在对该方程及若干相关不等式的认知上推进了数步。我们证明，在其无穷多解中存在一个满足椭圆正则性理论的特殊解。该解简化了Bogovskii常数$C_B$的定义，并使我们得以证明它在光滑区域中可达。随后我们得到了任意Lipschitz区域中$C_B$的通用下界，以及一个非极小性判据。正如预期，在区域变化时球体是极小值点，不过我们并未使用对称化技术。我们还分析了椭球和环域：针对前者，我们改进了目前针对薄化区域的最优渐近不等式。最后，我们引入了高阶Bogovskii常数，由此引出多调和Stokes问题。当源项具有某些零迹时，不仅正则性理论无需区域光滑性即可适用，我们还证明，在具有完全相同Bogovskii常数的Lipschitz区域中，球体同样是极小值点。文中提出了三个主要的挑战性开放问题。
英文摘要
The huge amount of literature about the divergence equation in bounded Lipschitz domains of $\R^n$ ($n\ge2$) is fairly disconnected and even apparently simple problems remain unsolved. We go several steps further in the knowledge of this equation and of some related inequalities. We prove that among its infinitely many solutions there exists a special one obeying elliptic regularity theory. This solution simplifies the definition of the Bogovskii constant $C_B$ and allows us to prove its attainment in smooth domains. We then obtain a universal lower bound for $C_B$ in any Lipschitz domain as well as a non-minimality criterion. As expected, balls are minimisers as the domain varies, although no symmetrisation technique is used. We also analyse ellipsoids and annuli: for the first we improve the (so far) best asymptotic inequality for thinning domains. Finally, we introduce higher-order Bogovskii constants, which lead to a polyharmonic Stokes problem. Not only regularity theory applies without smoothness of the domain when the source has some vanishing traces, but we also prove that balls are again minimisers among Lipschitz domains with the very same Bogovskii constant. Three main challenging open problems are suggested.
DOI：10.13140/RG.2.2.24025.89448
