Weak maximum principle for elliptic operators on a stratified set
Let \(\Omega \) be a connected set in \(\mathbb R^m\) which is the union of finitely many smooth manifolds (strata) \(\sigma_{ki}\) of different dimension. The subscript \(k\) is the dimension and the subscript \(i\) is the number of a stratum of this dimension. Strata regularly are adjacent to each other; if a stratum \(\sigma_{ki}\) is adjacent to \(\sigma_{k+1,j}\), then when a point \(y\in \sigma_{k+1,j}\) approaches a point \(x\in \sigma_{ki}\), the tangent space \(T_y \sigma_{k+1,j}\) tends to a certain limit position containing \(T_x \sigma_{k,i}\). The set \(\Omega \) has a cell complex structure. \(\Omega_0\) is a dense domain in \(\Omega \) composed of strata. For a nonnegative function \(q\) it is proved the weak maximum principle for a continuous function \(u\) on \(\Omega \) which solves the inequality \(L_q u \geq 0\) in \(\Omega_0\).
scientific article; zbMATH DE number 5319761
scientific article