Extension of functions from hypersurfaces with boundary

Abstract

A coretraction, a right inverse to the trace operator to a smooth hypersurface 𝒮 with the smooth boundary Γ = ∂𝒮 ≠ ∅, is constructed. The coretraction extends m-tuples of functions from the Besov spaces on each face of 𝒮 into the Bessel potential space on the domain slit by the hypersurface , provided these tuples satisfy a compatibility conditions on the boundary Γ. The traces are defined by arbitrary Dirichlet systems of boundary operators and extension is performed by two different methods, one implicit and one explicit. Explicit extension, based on the solution to the Dirichlet BVP for the poly-harmonic equation, permits the extension of distributions from the Besov space with a negative s < 0. Moreover, it permits to establish some additional features of the extended functions, which are useful in applications. Coretractions have essential applications in boundary value problems for partial differential equations when, for example, it is necessary to reduce a BVP with non-homogeneous boundary conditions to a BVP with the homogeneous boundary conditions.