AbstractThis chapter is devoted to the study of inhomogeneous boundary value problems. For this, we shall reformulate the boundary value problem again into a form which fits within the general framework of evolutionary equations. In order to have an idea of the type of boundary values which make sense to study, we start off with a section that deals with the boundary values of functions in the domain of the gradient operator defined on a half-space in $$\mathbb {R}^d$$ ℝ d (for d = 1 we have $$L_2(\mathbb {R}^{d-1})=\mathbb {K}$$ L 2 ( ℝ d − 1 ) = 𝕂 ).