On the Regularity of the Harmonic Green Potential in Nonsmooth Domains
Dorina Mitrea · Birkhäuser Boston eBooks · 2006
at least if f is well behaved. The mapping properties of Π are well understood. Essentially, this is a smoothing operator of order two on the scale of L-based Sobolev spaces H and this readily translates into mapping properties for the solution operator f → u = Πf of the partial differential equation (PDE) in question. Next, take the case when the Poisson problem is considered in a bounded domain Ω ⊂ R. This time, boundary conditions must be imposed and we consider the homogeneous Dirichlet boundary condition in which scenario the PDE reads Δu = f in Ω, Tru = 0 on ∂Ω, (15.4)