We are concerned with the location of the internal transition layer for some classes of solutions to an inhomogeneous one-dimensional elliptic problem with Neumann boundary conditions. We generalize and extend some known results with a variational method based on the Young's inequality and the co-area formula. As an application, we establish the location of the internal transition layer to the problem posed on some symmetric Riemannian manifolds.

中文翻译：

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在内部过渡层上，出现一些不均匀的半线性问题：接口位置

对于带有Neumann边界条件的非均匀一维椭圆问题的某些类型的解决方案，我们关注内部过渡层的位置。我们使用基于杨氏不等式和共面积公式的变分方法来概括和扩展一些已知的结果。作为一种应用，我们确定内部过渡层的位置，以解决某些对称黎曼流形上的问题。