SIAM Journal on Mathematical Analysis, Volume 53, Issue 2, Page 1745-1784, January 2021.
In a three-dimensional bounded possibly multiply connected domain, we prove the existence, uniqueness, and regularity of some vector potentials, associated with a divergence-free function and satisfying mixed boundary conditions. For such a construction, the fundamental tool is the characterization of the kernel which is related to the topology of the domain. We also give several estimates of vector fields via the operators div and curl when mixing tangential and normal components on the boundary. Furthermore, we establish some Inf-Sup conditions that are crucial in the $L^{p}$-theory proofs. Finally, we apply the obtained results to solve the Stokes problem with a pressure condition on some part of the boundary and Navier-type boundary condition on the remaining part, where weak and strong solutions are considered.

中文翻译：

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具有混合边界条件的矢量势。在压力和Navier型边界条件的斯托克斯问题中的应用

SIAM数学分析杂志，第53卷，第2期，第1745-1784页，2021年1月。
在三维有界的可能多重连接域中，我们证明了某些矢量势的存在，唯一性和正则性，与无散度函数相关并满足混合边界条件。对于这种构造，基本工具是内核的表征，该表征与域的拓扑有关。当在边界上混合切向分量和法向分量时，我们还通过运算符div和curl给出了矢量场的一些估计。此外，我们建立了一些Inf-Sup条件，这些条件在$ L ^ {p} $理论证明中至关重要。最后，我们将获得的结果应用到边界部分的压力条件和其余部分的Navier型边界条件的Stokes问题，其中考虑了弱解和强解。