A definition of the Laplacian on Cartesian products with mixed boundary conditions using quadratic forms is proposed. Its consistency with the standard definition for homogeneous and certain mixed boundary conditions is proved and, as a consequence, tensor representations of the corresponding Sobolev spaces of first order are derived. Moreover, a criterion for the domain to belong to the Sobolev space of second order is proved.

中文翻译：

---

具有混合边界条件的笛卡尔积上的拉普拉斯算子

提出了使用二次形式定义混合边界条件的笛卡尔积上的拉普拉斯算子的定义。证明了其与齐次和某些混合边界条件的标准定义的一致性，因此，得出了相应的一阶Sobolev空间的张量表示。此外，证明了该域属于二阶Sobolev空间的准则。