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Vector potentials in three-dimensional non-smooth domains. (English) Zbl 0914.35094

Considering curl as an operator in \(H_0(\text{div}): =\{\Phi\in L_2\mid \text{div} \Phi=0\}\), the \(L_2\)-orthogonal decomposition into the closure of the range of curl and its kernel space (the space of harmonic Dirichlet fields) becomes obvious. It is known that under rather weak assumptions a corresponding compact imbedding result holds which implies the closedness of the range and the finite-dimensionality of the kernel. Thus, in this case we have that elements \(\Phi\in H_0(\text{div})\) can be written as \(\Phi=\text{curl} \Psi\) with a so-called vector potential \(\Psi\in H_0(\text{div})\) if and only if \(\Phi\) is orthogonal to this finite-dimensional kernel.
In this paper, for the price of stricter regularity constraints on the boundary of the underlying 3-dimensional domain \(\Omega\) the latter orthogonality condition is characterized in terms of boundary traces. As a second vector potential construction fields \(\Phi\in H_0(\text{div})\) are considered, which are ‘tangential’ at the boundary. Here orthogonality to the corresponding kernel (the space of harmonic Neumann fields) is also characterized by trace conditions imposed on ‘cuts’ through the ‘handles’ of \(\Omega\). For the latter case also a third vector potential construction (requiring the boundary to be \(C^{1,1})\) is shown. The preference given to the trace characterization finds its proper justification in the discussion of the global regularity of the vector potential. This regularity is stated in terms of the Sobolev chain \((H_s(\Omega))_{s\in\mathbb{R}}\). In particular, the continuous imbedding into the Sobolev space \(H_1(\Omega)\), i.e. the well-known Gaffney inequality, is recovered for \(C^{1,1}\)-boundaries.
The results are utilized in a concluding chapter to construct divergence-free finite-element approximations. As a particular application a variational formulation of the Stokes problem is discussed.
Reviewer: R.Picard (Dresden)

MSC:

35Q30 Navier-Stokes equations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J20 Variational methods for second-order elliptic equations
76D07 Stokes and related (Oseen, etc.) flows
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