Half-Dirichlet problems for dirac operators in lipschitz domains

Abstract

Recall that in the case of the Dirichlet problem for the Laplace operator ∂2 x +∂ 2 y in Ω ⊆ R2, one prescribes the whole trace of a harmonic function in, say, L2(∂Ω). On the other hand, for the Cauchy-Riemann operator ∂x + i∂y, natural boundary problems are obtained by prescribing “half” of the trace of the analytic function in L2(∂Ω). Such half-Dirichlet problems arise when, for example, one prescribes the boundary values of the real part of an holomorphic function. This type of phenomenon has a natural extension to the higher dimensional setting in the context of Clifford algebras and Dirac operators. The main goal of this paper is to continue the line of investigation initiated in [Mi], [Mi2] (and further pursued in [McMM], [McM], [Mi3]) aimed at solving elliptic boundary problems in nonsmooth domains via tools originating in Clifford analysis. More specifically, here we shall study certain half-Dirichlet type problems for Dirac operators in nonsmooth Euclidean domains. Since no symbolic calculus for singular integral operators in the irregular context is available so far, one is forced to study such problems on an individual basis. Furthermore, it is well known that there are topological obstructions to the existence of local elliptic boundary conditions for general Dirac type operators (cf., e.g., [Gi]). Here we focus the discussion on the classical (full, as opposed to chiral) Dirac operator D := ∑ j ej∂j in Rm. Let Ω be an arbitrary Lipschitz subdomain of Rm. A specific example of a boundary value problem which fits the above description is