Using techniques of the theory of semigroups of linear operators, we study the question of approximating solutions to equations governing diffusion in thin layers separated by a semi-permeable membrane. We show that as thickness of the layers converges to 0, the solutions, which by nature are functions of 3 variables, gradually lose dependence on the vertical variable and thus may be regarded as functions of 2 variables. The limit equation describes diffusion on the lower and upper sides of a two-dimensional surface (the membrane) with jumps from one side to the other. The latter possibility is expressed as an additional term in the generator of the limit semigroup, and this term is built from permeability coefficients of the membrane featuring in the transmission conditions of the approximating equations (i.e., in the description of the domains of the generators of the approximating semigroups). We prove this convergence result in the spaces of square integrable and continuous functions, and study the way the choice of transmission conditions influences the limit.

使用线性算子的半群理论的技术，我们研究控制方程的近似解的问题，该方程控制由半透膜分隔的薄层中的扩散。我们表明，随着层的厚度收敛到0，本质上是3个变量的函数的解逐渐失去对垂直变量的依赖，因此可以视为2个变量的函数。极限方程描述了在二维表面（膜）的下侧和上侧的扩散，其中扩散是从一侧跳到另一侧。后者的可能性在极限半群的生成器中表示为一个附加项，并且该项是根据近似方程的传递条件（即，在近似半群的生成器域的描述中）。我们在平方可积和连续函数的空间中证明了这种收敛结果，并研究了传递条件的选择如何影响极限。