  We establish the existence and uniqueness, in bounded as well as unbounded
Lipschitz type cylinders of the forms $U_X\times V_{Y,t}$ and $\Omega\times
\mathbb R^{m}\times \mathbb R$, of weak solutions to Cauchy-Dirichlet problems
for the strongly degenerate parabolic operator \[ \mathcal{L}:=
\nabla_X\cdot(A(X,Y,t)\nabla_X)+X\cdot\nabla_Y-\partial_t, \] assuming that
$A=A(X,Y,t)=\{a_{i,j}(X,Y,t)\}$ is a real $m\times m$-matrix valued, measurable
function such that $A(X,Y,t)$ is symmetric, bounded and uniformly elliptic.
Subsequently we solve the continuous Dirichlet problem and establish the
representation of the solution using associated parabolic measures. The paper
is motivated, through our recent studies, arXiv:2012.03654, arXiv:2012.04278,
arXiv:2012.07446, by a growing need and interest to gain a deeper understanding
of the Dirichlet problem for the operator $\mathcal{L}$ in Lipschitz type
domains. The key idea underlying our results is to prove, along the lines of
Brezis and Ekeland, and in particular following the recent work of S. Armstrong
and J-C. Mourrat, arXiv:1902.04037, concerning variational methods for the
kinetic Fokker-Planck equation, that the solution can be obtained as the
minimizer of a uniformly convex functional.
