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We establish the existence and uniqueness, in bounded as well as unboundedLipschitz 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 problemsfor 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, measurablefunction such that $A(X,Y,t)$ is symmetric, bounded and uniformly elliptic.Subsequently we solve the continuous Dirichlet problem and establish therepresentation of the solution using associated parabolic measures. The paperis motivated, through our recent studies, arXiv:2012.03654, arXiv:2012.04278,arXiv:2012.07446, by a growing need and interest to gain a deeper understandingof the Dirichlet problem for the operator $\mathcal{L}$ in Lipschitz typedomains. The key idea underlying our results is to prove, along the lines ofBrezis and Ekeland, and in particular following the recent work of S. Armstrongand J-C. Mourrat, arXiv:1902.04037, concerning variational methods for thekinetic Fokker-Planck equation, that the solution can be obtained as theminimizer of a uniformly convex functional.

The Dirichlet problem for Kolmogorov-Fokker-Planck type equations with rough coefficients