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

We establish the existence and uniqueness, in bounded as well unbounded Lipschitz type cylinders of forms UX×VY,t Ω×Rm×R, weak solutions to Cauchy-Dirichlet problems for strongly degenerate parabolic operatorL:=∇X⋅(A(X,Y,t)∇X)+X⋅∇Y−∂t, assuming that A=A(X,Y,t)={ai,j(X,Y,t)} is a real m×m-matrix valued, measurable function such A(X,Y,t) symmetric, uniformly elliptic. Subsequently we solve continuous Dirichlet problem representation solution using associated measures. The paper motivated, through our recent studies [11], [12], [13], by growing need interest gain deeper understanding operator L domains. key idea underlying results prove, along lines Brezis Ekeland [3], [4], inspired work S. Armstrong J-C. Mourrat [2] concerning variational methods kinetic Fokker-Planck equation, can be obtained minimizer convex functional.