On the regularity of differential forms satisfying mixed boundary conditions in a class of Lipschitz domains

Abstract

Let Ω ⊂ ℝ n be a bounded Lipschitz domain, whose boundary decomposes into two disjoint pieces Σ t , Σ n ⊆ ∂Ω, which meet at an angle 0 with the property that if |2 - p| < e, then the following holds. Consider a vector field u with components u 1 ,..., u n ∈ L p (Ω) such that and curl u =(∂ j u k - ∂ k u j ) 1≤j,k≤n ∈ L p (Ω). Set ν · u = Σ n j =1 ν j u j and ν × u = (ν j u k - ν k u j ) 1≤j,k≤n . Then the following are equivalent: (i) (ν · u) |Σ t ∈ L p (Σ t ) and (v × u) |Σ n ∈ L p (Σ n ); (ii) ν · u ∈ L p (∂Ω); (iii) ν × u ∈ L p (∂Ω). Moreover, if either condition holds, then u belongs to the Besov space B p,max(p,2) 1/p (Ω). In fact, similar results are valid for differential forms of arbitrary degree. This generalizes earlier work dealing with the case when Σ t = O or Σ n = O.