## Boundary value problems for elliptic operators with singular drift terms

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28/11/2012##### Author

Kirsch, Josef

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##### Abstract

Let Ω be a Lipschitz domain in Rᴺ,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 ɛ L∞loc(Ω) is divergence free and satisfies the growth condition
dist(X,∂Ω)|B(X)|≤ ɛ1 for ɛ1 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 Lp(∂Ω), 1 < p < ∞, and the regularity problem for boundary
data in W¹,ᵖ(∂Ω) and HS¹.
The main result of this thesis is that the solvability of the regularity problem for boundary data
in HS1 implies the solvability of the adjoint Dirichlet problem for boundary data in Lᵖ'(∂Ω) and
the solvability of the regularity problem with boundary data in W¹,ᵖ(∂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 W¹,ᵖ(∂Ω) implies the solvability
with boundary data in HS1. 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 Lᵖ(∂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 ɛ L∞loc(Ω) such that in a small neighbourhood
of ∂Ω we have that dist(X,∂Ω)(|B(X)| + |C(X)| + |D(X)|) ≤ ɛ1 for ɛ1 small and that the
vector field B satisfies |∫B∇Ø| ≤ C∫|∇Ø| for all Ø ɛ Wₒ¹'¹ of that neighbourhood.