Relationship betwen $textbf{E}$, $textbf{D}$ and $U$ and boundary conditions

I am solving electric field $E$ inside an infinitely long tube, with inner radius a, and other radius b in a homogeneous electric field $textbf{E}_0$, as seen on the sketch. The pipe is from material with dielectric constant $epsilon$.

I tried to write potential $U$ as:

$$
U(r, phi)=sum_{m=1} ^infty (A_mr^m+B_mr^{-m})
$$

But I don’t, know how to set boundary conditions. I can assume, that I have non zero coefficients for A and B only for $m=1$, since otherwise I wouldn’t have constant potentials at border. From solution of problem I know that the boundary conditions are, that U is continuous at borders a and b and that $textbf{D}_r$, (strength of electric field in the direction of $textbf{D}_r$ ) is continuous.

But I don’t know from where this comes. So my question is, what are the connections between E, D and U, and how to calculate them. Especially, how to come to E inside the tube, once we calculate U. I know formula $D=epsilon epsilon_0 E$, but I don’t know when and whay to use D or E.