Boundary value problems in Lipschitz domains
for equations with drifts
In this work we establish solvability and uniqueness for the Dirichlet problem and the Regularity problem for second order elliptic operators in bounded Lipschitz domains, where is bounded, as well as their adjoint operators . The methods that we use are estimates on harmonic measure, and the method of layer potentials.
The nature of our techniques applied to for and for leads us to impose a specific size condition on in order to obtain solvability. On the other hand, we show that for and for are uniquely solvable, assuming only that is Lipschitz continuous (and not necessarily symmetric) and is bounded.
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