Weak equals strong for first-order linear pseudodifferential boundary value problems

Abstract:

Weak solutions are shown to be strong for a class of boundary value problems for first-order linear differential equations with zero order pseudodifferential matrix coefficients. The operators have the form $\chi {A_j}(x,D)\chi {D_j}\chi + \chi {A_0}\chi$, where $\chi$ is the characteristic function of a domain $G$. At the boundary, there are conditions on the coefficient ${A_n}$ of normal differentiation, typically implying that ${(\chi {A_n})^{ - 1}}$ is a bounded operator both on ${L_2}(G)$ and on ${H_1}(G)$. The proof uses tangential mollifiers and is also applied in a more abstract setting.