论文标题
具有粗糙系数的Kolmogorov-Fokker-Planck类型方程的Dirichlet问题
The Dirichlet problem for Kolmogorov-Fokker-Planck type equations with rough coefficients
论文作者
论文摘要
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 $Ω\times \mathbb R^{m}\times \mathbb R$, of weak solutions to Cauchy-Dirichlet problems for the strongly degenerate parabolic operator \[ \mathcal{L}:= \ \ nabla_x \ cdot（a（x，x，y，t）\ nabla_x）+x \ cdot \ nabla_y-- \ \ partial_t，\]，假设$ a = a = a（x，x，x，x，y，t）= \ {a_ {a_ {a_ {i，j}（i，j}（x，y，y，y，y，t） $ a（x，y，t）$是对称的，有界和均匀的椭圆形的。随后，我们解决了连续的Dirichlet问题，并使用相关的抛物线措施来建立解决方案的表示。通过我们最近的研究，ARXIV：2012.03654，ARXIV：2012.04278，ARXIV：2012.07446，通过越来越多的需求和兴趣，以对操作员$ \ \ \ \ \ Mathcal {L} $ in lipschitz Type域中的Dirichlet问题进行更深入的了解。我们结果的基础的关键思想是沿着布雷兹和埃克兰的界限，尤其是在阿姆斯特朗和J-C最近的工作之后。 Mourrat，Arxiv：1902.04037，关于动力学fokker-Planck方程的变异方法，可以将溶液作为均匀凸功能的最小化器获得。
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 $Ω\times \mathbb R^{m}\times \mathbb R$, 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.
