2020/12/21 by Malte Litsgård, Litsgård, M., Kaj Nyström +1
Computer Science · Mathematics · #35H20 #35K65 #35K70 #35R03 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2012.11410
openalex publication_date 2020/12/21 · openalex created_date 2021/01/05 · openalex updated_date 2026/07/28
We establish the existence and uniqueness, in bounded as well as unbounded Lipschitz type cylinders of the forms UX× VY,t and Ω× \mathbb Rm× \mathbb R, of weak solutions to Cauchy-Dirichlet problems for the strongly degenerate parabolic operator L:= ∇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 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 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.