Boundary value problems for elliptic operators with singular drift terms

Let Ω be a Lipschitz domain in R, n ≥ 3, and L = divA∇ − B∇ be a second order elliptic operator in divergence form with real coefficients such that A is a bounded elliptic matrix and the vector field B ∈ Lloc(Ω) is divergence free and satisfies the growth condition dist(X, ∂Ω)|B(X)| ≤ ε1 for ε1 small in a neighbourhood of ∂Ω. For these elliptic operators we will study on the basis of the theory for elliptic operators without drift terms the Dirichlet problem for boundary data in L(∂Ω), 1 < p < ∞, and the regularity problem for boundary data in W (∂Ω) and HS. The main result of this thesis is that the solvability of the regularity problem for boundary data in HS implies the solvability of the adjoint Dirichlet problem for boundary data in L ′ (∂Ω) and the solvability of the regularity problem with boundary data in W (∂Ω) for some 1 < p <∞. In [KP93] C.E. Kenig and J. Pipher have proven for elliptic operators without drift terms that the solvability of the regularity problem with boundary data in W (∂Ω) implies the solvability with boundary data in HS. Thus the result of C.E. Kenig and J. Pipher and our main result complement a result in [DKP10], where it was shown for elliptic operators without drift terms that the Dirichlet problem with boundary data in BMO is solvable if and only if it is solvable for boundary data in L(∂Ω) for some 1 < p <∞. In order to prove the main result we will prove for the elliptic operators L the existence of a Green’s function, the doubling property of the elliptic measure and a comparison principle for weak solutions, which are well known results for elliptic operators without drift terms. Moreover, the solvability of the continuous Dirichlet problem will be established for elliptic operators L = div(A∇+B)+C∇+D with B,C,D ∈ Lloc(Ω) such that in a small neighbourhood of ∂Ω we have that dist(X, ∂Ω)(|B(X)| + |C(X)| + |D(X)|) ≤ ε1 for ε1 small and that the vector field B satisfies | ́ B∇φ| ≤ C ́ |∇φ| for all φ ∈W 1,1 0 of that neighbourhood.