Hodge Decompositions on Weakly Lipschitz Domains

We survey the L₂ theory of boundary value problems for exterior and interior derivative operators $$ {d_{k1}} = d + {k_1}eo \wedge $$ and $$ {\delta _{k2}} = \delta + {k_2}eo $$ on a bounded, weakly Lipschitz domain $$\Omega \subset {{R}^{n}} $$, for k₁, k₂ ∈ C. The boundary conditions are that the field be either normal or tangential at the boundary. The well-posedness of these problems is related to a Hodge decomposition of the space L₂(Ω) corresponding to the operators d and δ In developing this relationship, we derive a theory of nilpotent operators in Hilbert space.Mathematics Subject Classification (2000). 35J55, 35Q60, 47B99.